The problem of the conflicting birth dates given by Matthew and Luke is mentioned in the wiki here: http://wiki.ironchariots.org/index.php? ... Birth_Date
The typical apologist's response to this is that Josephus doesn't tell us anything about what Quirinius was doing during the time that Matthew tells us Jesus was born (near the end of Herod's life). Therefore Quirinius must also have been governor of Syria at an earlier time, after all, he was governor many times in many places.
I've also heard it said that the census Luke describes (an "apographee" in Greek) must be a minor event that Josephus did not mention, as opposed to the later taxation ("phoron" in Greek) in 6 CE that caused a revolt and was therefore important enough for Josephus to record. Basically, Luke and Josephus are talking about entirely different events.
Obviously there is always some ambiguity involved in determining truth in history. But how true are these claims, and how much do they matter?
Perhaps the wiki should be expanded to include this information.
EDIT: Do we know who was governor of Syria during the last years of Herod's life? That would be an important piece of information.