The short answer is we aren't sure.
There are two basic methods that are commonly used to derive a birth year based on information from the Gospels.
The first is the nativity accounts in Luke and Mathew. Both reference the reign of Herod the Great. Since he died in 4 BC, that means it couldn't possibly have been any later than then. The main issue with this approach is that the nativity stories aren't generally accepted as historical. Worse yet, Luke also said it happened during the Census of Quirinius, which was 4 AD*.
The second is to take Luke 3:23's statement that Jesus was "about 30" when he started preaching, and try to work back from there. Combine this with some knowledge of the dates of the death of John the Baptist and the construction of the Great Temple and Jesus' visit there from John's Gospel, and you get around 1 BC. The big problem here is that there's a large chain of facts that have to be correct for this derived date to be accurate.
As for the actual day of year, the mention in Luke of shepherds out grazing their flocks implies that it could not have happened during the winter or late fall, but doesn't nail it down any better than that (and again, that's only if you accept Luke's nativity as historical).
* - This is a bit of an issue for those forced to work within a framework of Biblical Inerrancy. Fortunately as historians we are free to wield Occam's razor, which tells us one (at least) of these stated facts in Luke is in error.