This following verses in the Holy Qur'an in chapter 19 Maryam (Mary) describe with clarity the birth of Jesus (peace be upon him) and how that happen:

(Qur'an Surah Maryam (Mary) 19: 16 -> 36) :

<<And mention in the Book (the Holy Qur'an, O Mohammad, the story of) Mary, when she withdrew in seclusion from her family to a place facing east>>

<<She placed a screen (to screen herself) from them, then We sent to her Our Ruh (angel Gabriel), and he appeared before her in the form of a man in all respects.>>

<<She said " Verily! I seek refuge with the Most Beneficent (Allah) from you, if you do fear Allah>>

<<(the angel) said : " I am only a Messenger from your Lord, (to announce) to you the gift of a righteous son>>

<< She said:" How can I have a son, when no man has touched me, nor am I unchaste?>>

<<He said: "So (it will be), your Lord said: 'That is easy for Me (Allah): And (We wish) to appoint him as a sign to mankind and a mercy from Us (Allah), and it is a matter (already) decreed, (by Allah).' >>

<<So she conceived him, and she withdrew with him to a far place (i.e Bethlehem valley about 4-6 miles from Jerusalem).>>

<<And the pains of childbirth drove her to the trunk of a palm-tree. She said: " Would that I had died before this, and had been forgotten and out of sight!">>

<<Then (the babe "Jesus" or Gabriel) cried unto her from below her, saying: "Grieve not! Your Lord has provided a water stream under you; >>

<<And shake the trunk of palm-tree towards you, it will let fall fresh ripe-dates upon you">>

<<So eat and drink and be glad, and if you see any human being, say: "Verily! I have vowed a fast unto the Most Beneficent (Allah) so I shall not speak to any human being this day".>>

<<Then she brought him (the baby) to her people, carrying him. They said :" O Mary! Indeed you have brought a thing Fariyya (an unheard mighty thing)>>

<<O sister (i.e the like) of Aaron (not the brother of Moses, but he was another pious man at the time of Mary)! your father was not a man who used to commit adultery, nor your mother was an unchaste women. >>

<<Then she pointed to him. They said: " How can we talk to one who is a child in the cradle ? ">>

<<He (Jesus) said: Verily! I am a slave of Allah, He has given me the scripture and made me a prophet;>>

<<And He has made me blessed where-soever I be, and has enjoined on me prayer, and Zakat (charity), as long as I live.>>

<<And dutiful to my mother, and made me not arrogant, unblest.>>

<<And Salam (peace) be upon me the day I was born, and the day I die, and the day I shall be raised alive !>>

<<Such is Jesus, son of Mary. (it is) a statement of truth, about which they doubt (or dispute).>>

<<It befits not (the Majesty of ) Allah that He should beget a son (this refers to the slander of Christians against Allah, by saying that Jesus is the son of Allah). Glorified (and Exalted  be He above all that they associate with Him). When He decrees a thing, He only says to it, "Be"-and it is.>>

<<(Jesus said): "And Verily Allah is my Lord and your Lord. So worship Him (Alone). That is the Straight Path.">> (Allah's religion of Islamic Monotheism which He did ordain for all of his Prophets) [Tafsir At-Tabari].