Yes, there is indeed a connection with the Melchizedek legends. The birth story of Noah in 1 Enoch 106 occurs also in the Qumran Genesis Apocryphon (1Q19), which has survived in the original Hebrew. In 1 Enoch 106:5-6, Lamech tells his father Methuselah: "I have begotten a strange son, he is not like an ordinary human being, but he looks like the children of the angels of heaven to me; his form is different, and he is not like us. His eyes are like the rays of the sun, and his face glorious. It does not seem to me that he is of me, but of angels". In the Genesis Apocryphon, we also that Lamech "decided that the conception was at the hands of the Watchers, that the seed had been planted by the Holy Ones or Nephilim" (3:1), but his wife Bitenosh told him: "This seed comes from you, this conception was by you, the planting of this fruit is yours, not by any stranger, neither by any Watcher, nor yet by any of the Sons of Heaven" (3:15-16). There is a common motif here with the Jesus nativity story of Matthew, in that Joseph (like Lamech) recognized that the blessed child was not his and (implicitly) suspected that someone else fathered the child.
In the later (second century AD) book of 2 Enoch, the story was shifted from Lamech to Noah's brother Nir, and the "strange son" was Melchizedek. In 2 Enoch 71-72, we read that Nir's wife Sothonim gave birth to a wonder child "in the time of her old age and on the day of her death" (i.e. she was barren like Sarah in Genesis), even though her husband "had not slept with her, nor had touched her" (71:2). The combined motif is barrenness + conception without sexual intercourse. The latter component is present in the Lukan+Matthean narrative on Mary, while the former is present in the Lukan narrative on Elizabeth (presumably originally applied to Mary). Moreover, Zechariah (like Nir) was a "priest", and Elizabeth was "well along in years" in the Lukan story (Luke 1:5-7). Next, the text says that Sopanim "hid herself during all the days until she gave birth" (2 Enoch 71:3). This motif of isolation of the mother appears in the Ascension of Isaiah regarding Mary:
"He did not live with her for two months. And after two months of days while Joseph was in his house, and Mary his wife, but both alone" (Ascension of Isaiah 11:5-7; compare Matthew 1:25, "He had no union with her until she gave birth to a son").
When Nir found out that his wife was pregnant, he told her to "depart from me" (72:6), just as Joseph tried to "put [Mary] away" in the Ascension of Isaiah, but Sopanim replied in her defense: "I do not understand how my menopause and the barrenness of my womb have been reversed" (72:7). Then she died on the spot, and Nir became very upset. Then who should show up to console him but the archangel Gabriel:
"The archangel Gabriel appeared to Nir, and said to him: 'Do not think that your wife Sopanim has died because of your error; but this child which is to be born of her is a righteous fruit, and one whom I shall receive into paradise, so that you will not be the father of a gift of God" (2 Enoch 72:11).
This is strikingly reminiscent of Matthew 1:20, wherein an "angel of the Lord" appears to Joseph in a dream to tell him that Mary's son "is from the Holy Spirit" rather than of human parentage and will be destined to "save his people from their sins". In Luke, the resemblance is even closer, wherein the angel is named Gabriel and appears to Mary to tell her that "the Holy Spirit will come upon you, and the power of the Most High will overshadow you," and child will be a "holy one ... called the Son of God" (Luke 1:35). Then Nir and Noah go to bury Sopanim, and while they are digging her grave, "a child came out from the dead Soponim" who was as "strange" as the son of Lamech in 1 Enoch:
"They saw the child sitting beside the corpse, and having clothing on him. And Noah and Nir were very terrified, because the child was fully developed physically, like a three-year-old. And he spoke with his lips, and he blessed the Lord. And Noah and Nir looked at him closely, saying, 'This is from the Lord, my brother.' And behold, the badge of priesthood was on his chest, and it was glorious in appearance" (2 Enoch 71:17-19).
The theme of precociousness appears in 1 Enoch as well, and while it is missing in the canonical gospels it also appears in the Infancy Gospel of Thomas and other Christian legends of the young Jesus. Then Noah and Nir feed the child some "holy bread" and name him Melchizedek. Then, later on in the chapter, the archangel Michael takes Melchizedek to heaven to safety in Paradise so he would not perish during the Flood.
Since Melchizedek appears as a messianic figure in 11Q13 and is explicitly linked with Jesus in Hebrews 5:6, 10; 6:20; 7:1-17, the connection between the birth legend of Melchizedek and Jesus is especially important. In fact, there is direct evidence that this birth legend was applied to Jesus in Christian tradition. F. I. Andersen, in his translation of 2 Enoch, notes that "in a Christian legend, when Jesus goes to school and confounds his teacher Levi with his erudition, Levi exclaims, 'I think he must have been born before the Flood, before the Deluge' " (cf. Luke 2:46-52; Infancy Thomas 6:1-8:2; 15:1-4). This is almost certainly an allusion either to Enoch or Melchizedek. The resemblance between the stories however is also especially close in the case of Mary's conception in Infancy James (cf. especially in Joachim being a priest, Anna as old but childless, and Joseph's severe reproach on Mary for her pregnancy).
There are a few other features in the Melchizedek story in 2 Enoch that resembles legends about Jesus. In the Quran, the infant Jesus gives an oration from his cradle shortly after being born (Sura 19), and the same story is told in the older (but still comparatively late) Arabic Gospel of the Infancy. Also, the child Melchizedek is "entrusted to the care" of the archangel Michael in 2 Enoch 72:3. This evokes a fragment of the Gospel of the Hebrews, cited by Cyril of Alexandria (Discourse on Mary Theotokos 12a), which claimed that "when Christ wished to come upon the earth to men, the good Father summoned a mighty power in heaven, which was called Michael, and entursted Christ to the care thereof. And the power came into the world and it was called Mary, and Christ was in her womb seven months". Then, when Michael is about to take Melchizedek into Paradise, the text says that "Michael took the child on the same night on which he had come down; and he took him on his wings, and he placed him into the paradise of Edom [=Eden]" (2 Enoch 72:9). Interestingly, Origen cites another fragment of the Gospel of the Hebrews wherein Jesus says: "Even so did my mother, the Holy Spirit, take me by one of my hairs and carry me away onto the great mountain Tabor" (Commentary on John, 2.12.87). The concept is similar to that in 2 Enoch (where the one flying Melchizedek to heaven is Michael) and Ezekiel 8:3 which refers to an angel who "took me by a lock of my head; and the Spirit lifted me up to between earth and heaven".
The Jesus tradition, thus, appears to be quite dependent in some way on the cluster of motifs attested in the legend of Melchizedek and the birth of Noah in 1 Enoch. Indeed, when we take into account the haggada based on the birth of Moses, the material from 1 Samuel, Nazorean traditions of Miryai, the bar-Joseph and Joshua messianic titles, the star messianic motif from midrash on Numbers, and contemporary mother-goddess concepts from Hellenistic mysteries, is there any major feature in the nativity stories of Matthew and Luke that could not be traced to plausible antecedent traditions?