I'm not a historian but here is an excerpt from the Bible Knowledge Commentary...
a. Jesus' birth (2:1-7)
2:1-2.
Jesus' birth was dated by Luke as falling in the reign of Caesar Augustus, who was officially made the ruler of the Roman Empire in 27 b.c. and ruled to a.d. 14. (See the list of Roman emperors.) Because Herod the Great's reign ended in 4 b.c., Jesus was born before that time. The mention of Quirinius as governor of Syria poses a problem. He was governor in a.d. 6-7, much too late for Jesus' birth. Therefore does the word first hprôtê refer, as in the niv, to a first, that is, an earlier, census by Quirinius? If so, one would have to posit a previous governorship for Quirinius at about 4 b.c. Perhaps a better solution is to take "first" to mean "before," as it does, for example, in John 15:18. Luke 2:2 would then read, "this was the census that took place before Quirinius was governor of Syria" (i.e., before a.d. 6). (Bible Knowledge Commentary)