My understanding of this is:
There was a census by Quirinius, who was “governor of Syria”, in 6AD. This was famous because there were incidents and uprisings caused by the ill will of the people towards this.
Clearly, this census was much too late to be the one Luke refers to, as scholarship has well fixed the date of the death of Herod the Great in 4BC and scripture is clear Our Lord was born before his death.
The odd part here is that Luke is a superb historian, and in both his Gospel and in Acts is very, very careful to provide historical references, which check out with extra-biblical sources to a very high degree.
So it seems the key, as NotWorthy has pointed out, is that St. Luke refers to this as the “First census”, clearly implying more than one.
Also note that St. Luke doesn’t say that Quirinius **called **for the census, just that it took place during his reign as “Governor of Syria”; its not clear if it was actually called for by Caesar (which would probably be recorded somewhere but isn’t) or by someone acting in his name as a local ruler.
This is problematic in that Quirinius, to the best of our knowledge, wasn’t governor of Syria in 4BC. He was in the region as a military leader, and had been sent by Rome at that time to wipe out a rebellion in the area. But another man was Governor - I cant find my notes on that right now, but Im pretty sure Im right.
So the best explanation to square with Luke for me would be that a census was called in 4BC, by someone, in the name of Caesar. At that time, Quirinius, as a powerful military leader, might have held a “governorship” over some Syrian lands he was busy subduing. Luke refers to this, and to highlight the differences, is careful to say the “first census” to differentiate it from the more famous census of 6AD.
It is a mystery tho, and one that folks argue about, no doubt.