In the Qur'an and other Islamic sources, there is no mention of Joseph the carpenter, nor any recollection of the inn and manger legend. On the contrary, the Qur'an describes that Mary retreated from her people (outside the city), and gave birth to Jesus underneath a remote date palm tree. The tree miraculously provided nourishment for her during labor and birth. (See Chapter 19 of the Qur'an for the entire story. The chapter has aptly been named "The Chapter of Mary.")
However, the Qur'an repeatedly reminds us that Adam, the first human being, was born with neither a human mother nor a human father. Therefore, Jesus' miraculous birth affords him no higher standing or presumed partnership with God. When God ordains a matter, He merely says, "Be" and it is so. "The similitude of Jesus before God is as that of Adam. He created him from dust, then said to him: "Be!" And he was" (3:59). In Islam, Jesus is regarded as a human prophet and messenger of God, not part of God Himself.
Muslims observe two holidays per year, which are associated with major religious observances (fasting and pilgrimage). They do not revolve around the life or death of any human being, including prophets. While some Muslims observe the Prophet Muhammad's birthday, this practice is not universally accepted among Muslims. Therefore, most Muslims do not find it acceptable to celebrate or acknowledge the "birthday" of Jesus either.