I'm not going to stick around here too often, but this time of the year this same question comes up, and I'm tempted to answer. First of all, as Clark Thompson stated, the bible is silent on it (at least from a historical mention). If God had wanted us to know, he would have clearly had it written. Perhaps he did not because most of the masses of people would then get caught up on birthdays and such celebrations which the bible tends to frown upon. But the answer is there for those few who are deep enough to search.
I have several reasons why I believe it's in September:
1) The book of Luke mentions the priestly course of Abijah. Thru logical deduction of this timing, biblical historians have deduced (correctly IMO) that he would have been born late summer or early autumn.
2) The book of Luke states that Jesus began his ministry at (about) 30 years of age. Also, there is a passage where Jesus compares his ministry to 3.5 years of Elijah (also consider in the end times the 3.5 years, or more specifically 1260 days, of the 2 witnesses). This made him 33.5 years old when he was crucified at the Spring feasts, thus born in the autumn around September time frame.
3) Revelation 12, amidst it's symbolism, also is suggestive of something else that most overlook: an astronomical position. There's only one zodiac sign that's a woman that the sun travels thru, and that's Virgo. The woman (Virgo) is clothed with the sun, with the moon under her feet (near to Libra). For the woman to be "clothed", the sun in that epoch had to be in Virgo
mid September. Anything later than that would mean the sun is at her feet, and as a gentleman I would not consider a woman clothed with knee-high stockings and I would turn my head.
Likewise, when the moon is under her feet only happens at one time:
The Feast of Trumpets. Jews today call that Rosh Hoshana. This feast is prophetic and indicative of the coming and coronation of a King. In the end times that's when Christ will return.
Now that's all you need to know without me going into even greater detail, although I believe that Christ lived exactly 12240 days (34 x 360) which are 34 prophetic years (and that also works out to 33.5 years) which comes from yet another source. Adding this to a mid-September date always works out to a late March crucifixion day, and by calculating back to a candidate year to the day of the week I know he was crucified, I have derived very precise birth/death dates which also agree with history. Of course people can dispute my findings, but given the criteria, nothing else mathematically works out, therefore to circumvent my dates one basically has to claim the criteria are not valid or biblical and sweep them under the rug hoping they go away.