Matthew tells us that Jesus was born before the death of King Herod, which we know from history was in 4 BC. On the other hand, Luke appears to tell us that Jesus was born while Quirinius was ruling Syria, which he started to do in 6 AD.
So the order of events is as follows:
| Matthew 2:1 | ... Jesus is born |
| Matthew 2:19 | ... Herod dies in 4 BC |
| Luke 2:1-5 | ... Mary and Joseph go to Bethlehem after Quirinius becomes Syrian governer in 6 AD |
| Luke 2:6-7 | ... Jesus is born |
However, it may be possible to avoid this problem with a different translation of Luke 2:2. Here is the normal translation:
The ESV's notes give as an alternate translation:
This could reconcile the dates, as it would mean Quirinius was not yet governor. Only a competent scholar of ancient Greek could say whether this translation is legitimate. It's worth noting that evangelical Bibles like the ESV, NASB and even the NIV see the problematic translation as the more probable one.
There's also the following question: why would Luke mention Quirinius, who would only become ruler ten years later? I can think of one reason; if there was a second census at a later time, Luke might want to distinguish between the two. The census of 6 AD is fairly well attested - indeed, a man called Judas of Galilee led a rebellion in protest at it. But was there a prior census around 4 BC? I don't know.