From my own knowledge, there has never been any direct reference to when Jesus was born in any of the Abrahamic religious works.
The Qur'an tells the story of Jesus' birth mainly in 3:45-51 but there are no clues as to the date or even time of year this event took place.
Likewise in the Christian Bible, in the Gospels. The only clue we get is that it was during the time of King Herod who was the Roman king of Judea between 37 BCE– c. 4 BCE .
There are many theories about the time of year of Jesus' birth, but referencing the Qur'an we get our biggest clue.
After Mary gave birth the pain drove her to a date tree of which she shook the trunk....
“Grieve not! Your Lord has provided you a stream of clear water 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...” (Quran 19:24)
Dates are usually harvested in September, and add to this fact the the shepherds were still in the fields tending their sheep as the cold weather hadn't drawn in yet gives us another little pointer.
Islam, of course doesn't celebrate the birth of Jesus as they do not believe in the Trinity but do hold both Jesus and Mary in very high regard as they do all prophets and their mothers.
Great question, hope this helps :-)