Although nothing that Jesus actually said or taught in the Gospels is reiterated in the Quran, the Quran does recount stories of Jesus.
Firstly, He is said to have been born when Mary was alone in the wilderness, under a palm tree, shortly after which He speaks in defense of His mother, as an infant in the cradle. Then, He is also said to have fashioned birds from clay, which He then imbued with life:
Surah 19:22-33
22 So ˹Mary˺ conceived him and withdrew with him to a remote place.
23 Then the pains of labour drove her to the trunk of a palm tree. She cried, "Alas! I wish I had died before this, and was a thing long forgotten!"
24 So a voice reassured her from below her, "Do not grieve! Your Lord has provided a stream at your feet.
25 And ˹if you˺ shake the trunk of this palm tree towards you, it will drop fresh, ripe dates upon you.
26 So eat and drink, and put your heart at ease. But if you see any of the people, say, 'I have vowed silence to the Most Compassionate, so I am not talking to anyone today.'"
27 Then she returned to her people, carrying him. They said ˹in shock˺, "O Mary! You have certainly done a horrible thing!
28 O sister of Aaron! Your father was not an indecent man, nor was your mother unchaste."
29 So ˹Mary˺ pointed to the baby. They exclaimed, "How can we talk to someone who is an infant in the cradle?"
30 Jesus declared, "I am truly a servant of Allah. He has destined me to be given the Scripture and to be a prophet.
31 He has made me a blessing wherever I go, and bid me to establish prayer and give alms-tax as long as I live,
32 and to be kind to my mother. He has not made me arrogant or defiant.
33 Peace be upon me the day I was born, the day I die, and the day I will be raised back to life!"
Surah 3:48-49
48 And Allah will teach ˹Jesus˺ writing and wisdom, the Torah and the Gospel,
49 and ˹make him˺ a messenger to the Children of Israel ˹to proclaim,˺ "I have come to you with a sign from your Lord: I will make for you a bird from clay, breathe into it, and it will become a ˹real˺ bird - by Allah's Will.
Surah 5:110
110 And ˹on Judgment Day˺ Allah will say, "O Jesus, son of Mary! Remember My favour upon you and your mother: how I supported you with the holy spirit so you spoke to people in ˹your˺ infancy and adulthood. How I taught you writing, wisdom, the Torah, and the Gospel. How you moulded a bird from clay - by My Will - and breathed into it and it became a ˹real˺ bird - by My Will.
While these stories have no basis in the Bible, they do have a basis in pseudepigraphal scriptures written hundreds of years after the time of Jesus.
Before examining these sources, it is essential to state what no one is claiming. No one is claiming that a word-for-word copy-paste took place, between a written source and the Quran, in any of these instances. Rather, we are asserting that the reason these narratives are in the Quran is a result of Muhammad's oral familiarity with local legends, which had their source in heretical, forged texts, like the ones that will be examined. Also, although legends can exist in oral form in many languages prior to their being written down, in any language, the dates for the texts which will be covered here are all well within a timeframe that would implicate Muhammad, often predating him by many hundreds of years.
Firstly, the account of Mary giving birth under a palm tree (Surah 19:22-27) seems to have been derived from the Gospel of Psuedo-Matthew, a late source, dated to around the early 7th century, which says:
Gospel of Psuedo-Matthew Chapter 20
20 And it came to pass on the third day of their journey, while they were walking, that the blessed Mary was fatigued by the excessive heat of the sun in the desert; and seeing a palm tree, she said to Joseph: Let me rest a little under the shade of this tree. Joseph therefore made haste, and led her to the palm, and made her come down from her beast. And as the blessed Mary was sitting there, she looked up to the foliage of the palm, and saw it full of fruit, and said to Joseph: I wish it were possible to get some of the fruit of this palm. And Joseph said to her: I wonder that you say this, when you see how high the palm tree is; and that you think of eating of its fruit. I am thinking more of the want of water, because the skins are now empty, and we have none wherewith to refresh ourselves and our cattle.
Then the child Jesus, with a joyful countenance, reposing in the bosom of His mother, said to the palm: O tree, bend your branches, and refresh my mother with your fruit. And immediately at these words the palm bent its top down to the very feet of the blessed Mary; and they gathered from it fruit, with which they were all refreshed. And after they had gathered all its fruit, it remained bent down, waiting the order to rise from Him who had commanded it to stoop. Then Jesus said to it: Raise yourself, O palm tree, and be strong, and be the companion of my trees, which are in the paradise of my Father; and open from your roots a vein of water which has been hid in the earth, and let the waters flow, so that we may be satisfied from you. And it rose up immediately, and at its root there began to come forth a spring of water exceedingly clear and cool and sparkling. And when they saw the spring of water, they rejoiced with great joy, and were satisfied, themselves and all their cattle and their beasts. Wherefore they gave thanks to God.
The above has the palm tree giving them fruit, and also a stream appearing at the tree, two details that match exactly what is seen in Surah 19:22-27 - though this has Jesus already having been born when Mary sought refuge under the tree, whereas the Quran has Mary actually give birth under the tree. And of course, it should be mentioned that the Quran's narrative of Mary going into the desert alone, and returning with the baby Jesus, completely contradicts the Gospels' account, which says that He was born with Joseph present, in a crowded Bethlehem (Matthew 2:1-14, Luke 2:1-7), just as prophesied (Micah 5:2) - something that even the ones forging Pseudo-Matthew would have been familiar with, yet Muhammad apparently was not.
Secondly, the account of Jesus speaking from the cradle (Surah 19:28-33, 5:110) comes from the Arabic/Syriac Infancy Gospel, which thought to be a later compilation of elements from the Infancy Gospel of Thomas, the Infancy Gospel of James, and the already-covered Gospel of Pseudo-Matthew.
In it, Jesus speaks from the cradle, exactly as is described in Surah 19:29 and 5:110:
Arabic Infancy Gospel 1:1
1 We find what follows in the book of Joseph the high priest, who lived in the time of Christ. Some say that he is Caiaphas. He has said that Jesus spoke, and, indeed, when He was lying in His cradle said to Mary His mother: I am Jesus, the Son of God, the Logos, whom thou hast brought forth, as the Angel Gabriel announced to thee; and my Father has sent me for the salvation of the world.
Finally, the source of the Quran's story of Jesus creating clay birds, and then bringing them to life (Surah 3:48-49, 5:110), comes directly from the aforementioned Infancy Gospel of Thomas, which is different from the well-known Gospel of Thomas - likewise a Gnostic apocryphal text - and is thought to date from the late 2nd century.
In it, a story is told of Jesus fashioning birds out of clay, and giving them life, just as He does in the Quran:
Infancy Gospel of Thomas 2:1-5
1 This little child Jesus when he was five years old was playing at the ford of a brook: and he gathered together the waters that flowed there into pools, and made them straightway clean, and commanded them by his word alone.
2 And having made soft clay, he fashioned thereof twelve sparrows. And it was the Sabbath when he did these things (or made them). And there were also many other little children playing with him.
3 And a certain Jew when he saw what Jesus did, playing upon the Sabbath day, departed straightway and told his father Joseph: Lo, thy child is at the brook, and he hath taken clay and fashioned twelve little birds, and hath polluted the Sabbath day.
4 And Joseph came to the place and saw: and cried out to him, saying: Wherefore doest thou these things on the Sabbath, which it is not lawful to do? But Jesus clapped his hands together and cried out to the sparrows and said to them: Go! and the sparrows took their flight and went away chirping.
5 And when the Jews saw it they were amazed, and departed and told their chief men that which they had seen Jesus do.
The above citations demonstrate that the non-Biblical accounts about Jesus's infancy that made it into the Quran have their roots in pseudepigraphal scriptures, forged in the names of people who died long before they were written, which have never been accepted by any mainstream group of Christians as authoritative or inspired Scripture. These scriptures also contain things that Muslims reject, and teach the Deity and Sonship of Jesus Christ, meaning that they definitely offer no evidence for a genuinely "Muslim Jesus".
Furthermore, it is known why these infancy Gospels were written. The four divinely-inspired Gospels which Christians have always accepted say almost nothing of Jesus Christ's life from His birth to the age of 12, when a single narrative is given in Luke. Therefore, deceivers who were not content with God's silence as to this period of the Lord's life saw their opportunity to fill in the gaps, using the names of people they had never met, in order to teach whatever strange doctrine they were desiring to teach.
Also, the stories highlighted here, in particular, are attempts to make the baby Jesus seem less fully human and "more" overtly divine, which not only runs contrary to the message of Islam, but also to the witness of inspired Scripture, which says that Jesus allowed himself to "increase in wisdom and stature" (Luke 2:52), growing up as a normal child. After waiting for the forerunner John the Baptist to announce the Kingdom of God (Matthew 3:1-3, etc.), He began His ministry at about age thirty (Luke 3:23), and that is when He started performing miracles (John 2:11). That is why his parents are surprised by His behavior in Luke 2:40-52 - an account which would make no sense if Jesus was walking around Nazareth prolifically performing miracles as a child. Similarly, passages like Matthew 13:54-57, and Mark 6:1-4 would make no sense if Jesus was renowned as a miracle-worker prior to His ministry.
Rather than affirming the Gospel, as he claimed (Surah 3:3-4, 5:46-47, 5:68, etc.), Muhammad repeated nothing that Jesus actually taught, instead repeating fables about Jesus Christ that were circulating in his locality in 7th century Arabia, which have never been accepted as inspired Scripture, and teach things contrary to accepted, inspired Scripture.
The Quran also mentions Mary living in the Temple in Jerusalem, and miraculously receiving food there:
Surah 3:37
37 So her Lord accepted her graciously and blessed her with a pleasant upbringing - entrusting her to the care of Zachariah. Whenever Zachariah visited her in the sanctuary, he found her supplied with provisions. He exclaimed, "O Mary! Where did this come from?" She replied, "It is from Allah. Surely Allah provides for whoever He wills without limit."
This comes from the early apocryphal text called the Protoevangelium of James, wherein Mary lives in the Temple, is attended to by Zechariah the priest, and is fed by angels:
Protoevangelium of James 8:1-3
1 And her parents went down, marveling and praising and glorifying the Lord God that the child hadn't turned back. And Mary was in the Temple of the Lord. She was nurtured like a dove, and received food from the hand of an angel.
2 And when she became twelve years old, there was a council of the priests, saying, "Look, Mary has been in the Temple of the Lord twelve years. What should we do about her so that she won't pollute the sanctuary of the Lord our God?" And they said to the chief priest, "You stand at the altar of the Lord. Go in and pray about her, and if the Lord God reveals anything to you, we'll do it."
3 And the chief priest went in, taking the robe with twelve bells into the Holy of Holies, and prayed about her. And look! An angel of the Lord stood nearby, saying, "Zechariah, Zechariah, go out and assemble the widowers of the people, and let them each bear a staff. And whomever the Lord God points out with a sign, she'll be his wife."
Protoevangelium of James 13:2
2 And Joseph stood from the sackcloth and called her and said to her, "God cared for you. Why have you done this? You've forgotten the Lord your God. Why have you humiliated your soul? You were nourished in the Holy of Holies and received food from the hand of an angel!"
Protoevangelium of James 15:3
3 And the chief priest said to her, "Mary, why have you done this? Why have you humiliated your soul and forgotten the Lord your God? You were raised in the Holy of Holies, and received food from the hand of an angel, and you heard its hymns and danced before it. What is this that you've done?"
For a refutation of the Protoevangelium of James - one of the most influential forgeries of all time - see this article.