"This was the first census that took place while Quirinius was governor of Syria."I think that this is a good translation (NIV) of the Greek, but the ambiguity comes from the Greek word prote meaning "first." Luke's use of "first" opens the door to the possibility that Quirinius had conducted a "first" census prior to the one in 6 C.E., thus reconciling Luke's timeline. While it might be desirable to reconcile Luke's chronology, I do not think that an earlier census makes sense of the data. Sure, Roman censuses had taken place before 6 C.E., and there is positive evidence of these censuses, none of them refer to a census of Judea during the reign of Herod the Great. The most common appeal that I have seen is to a census in 8 B.C.E. Yet, this was a census in Egypt, not in Judea. The main line of reasoning that precludes a census in Judea during the reign of Herod the Great is the fact that a census for the purpose of taxation would not have been needed in Judea as long as Herod, who was a good administrator if nothing else, was collecting and submitting taxes to Rome. And there is no positive evidence that he was not. In fact, if Herod were not doing his job, he would have been deposed, as was his son Archelaus 10 years after Herod's death. So, it is historically implausible that Rome conducted a census in Judea during the reign of Herod the Great. I concur with the statement of the late Raymond Brown where he states:
"There is no serious reason to believe that there was a Roman census of Palestine under Quirinius during the reign of Herod the Great. (Indeed, as regards the non-biblical 'evidence,' it is doubtful that anyone would have even thought about an earlier census if he [sic] were not trying to defend Lucan accuracy.)"*All of this reasoning leads to the conclusion that for whatever reason (and there could be many), Luke has not provided accurate historical information regarding the birth of Jesus. Either Jesus was born during the reign of Herod the Great (before 4 B.C.E), or during the census of Quirinius (6 C.E.), but it could not be during both.
* Raymond Brown, The Birth of the Messiah, (Garden City, NY: Doubleday, 1979), 554.