This is going to be very brief. For those who really want to get to the bottom of this I give two free online articles at the end.
The argument of Andrew Steinmann, John Rhoads and others is that Josephus got himself into a bit of a mess, and misunderstood his sources. It isn't just that Josephus disagrees with Luke's data in Scripture, it is that Josephus disagrees with Josephus.
For instance, Josephus says that the High Priest Joazar was made High Priest by Herod the Great because he opposed a certain "Judas" who was trying to persuade the Jews not to cooperate with the Roman census. Josephus then tells us in one place that Joazar was deposed by Archelaus in the days immediately after the death of Herod the Great to try to appease the Jews. But in another section Josephus tells us Cyrenius deposed Joazar about 7 AD: it cannot be both.
Josephus speaks of three men called Judas, he speaks of them as three different people - Steinmann et al argue these three Judases are actually the same man, who led an insurrection in the days of the census which actually occurred in the reign of Herod the Great.
The comments of Josephus that there was an insurrection by "a Judas of Galilee" in the days of the census is confirmed by Acts 5:37:
"After this man rose up Judas of Galilee in the days of the taxing [census], etc..."
The census being referred to in Acts 5:37 is believed to be the census which brought Joseph and Mary to Bethlehem.
[[[[ Rambling Diversion for those who are interested !
(It is not surprising that Joseph knew that Bethlehem was his ancestral home, even though King David was a thousand years before... Joseph was the direct descendant of David, and if the Davidic dynasty continued up to Joseph, then Joseph himself would have been the King. Furthermore, the line of descent of the males was recorded by the priesthood in Jerusalem, probably when each baby boy came for circumcision. It would have been these legally binding records which Matthew and Luke would have consulted for their genealogies of Christ.)
It needs to be pointed out that in English translations there are different renderings of Luke 2:2.
The NIV says "This was the first census that took place while Quirinius was governor of Syria"; the KJV "this taxing was first made when Cyrenius was governor of Syria". It is believed that both of these versions have problems: the NIV implies that there was more than one census over a relatively short space of history; this seems unlikely due to the cost, and the method of taking the census seems designed to make another in such a short space of time unnecessary.
The KJV speaks of a taxing rather than a census.. it may be true that the census was mostly taken in order to help with taxation, but it wasn't a taxing in itself, it was merely a census to assist with future tax gathering. (It is very obvious that no one would go to Bethlehem, away from the evidence of their wealth at home in Nazareth, in order to have their wealth assessed!) No, the only purpose of the travel to Bethlehem is to have recorded their names and addresses. In a largely agricultural empire are the census makers going to arduously travel around to every hamlet, every remote homestead and single hovel to record every name? No, let the census authorities set themselves down/set up shop in the towns and let the people in the countryside around come to them. There aren't enough census makers to take any other policy; any other method would take up decades of time.
Why not just take a census in the hometown (eg Nazareth) rather than the ancestral town (Bethlehem)? Maybe:
First, that was probably not how Roman citizens were already recorded, they were recorded in the ancestral town. For instance, the Apostle Paul could wander around the Empire and simply tell people he was a Roman citizen to create genuine consternation in any official who mistreated him. Where was the evidence he was a Roman citizen? It would have been in the public records office in Tarsus.
Second, recording in the hometown creates problems if people move to a different town after the census. Nomatter how many times a person moves to a different town, their ancestral town remains the same.
Finally, both the KJV and the NIV say the activity was performed while Quirinius (Greek form)/ Cyrenius (Latin) "was Governor of Syria". I am told that the Greek does not actually say that and that a better translation of Luke 2:2 is:
"This census first took place while Quirinius was governing Syria" (NKJV).
What is the difference? Quirinius was not the Governor of Syria until 6 AD, but may have been acting as governor without the formal title previous to 6 AD.
End of First Rambling Diversion ]]]]
{{ Second Rambling Diversion On the Lunar Eclipses
Since comments have been made by others concerning a lunar eclipse let me briefly outline the issue. Josephus says that shortly before Herod the Great fell ill leading to his death there was a lunar eclipse. Josephus also indicates that after the eclipse a number of (unsuccessful) attempts to heal Herod were made, he died, had a very grand funeral to which many from far and wide were invited, and then Archelaus was acting ruler (needing confirmation from the Roman Emperor) when the Passover took place. There are four contending dates for the lunar eclipse, to which I add the number of days before Passover:
23 March, 5 BC, 29 days, total eclipse;
15 Sept, 5 BC, 7 months, total eclipse;
13 March, 4 BC, 29 days, partial eclipse;
10 Jan, 1 BC, 89 days, total eclipse.
All other eclipses mentioned on NASA website for the period can be ignored because they were not visible from Jerusalem, e.g. they happened during daylight hours in Jerusalem.
A discussion of the impossibility of squeezing all the events included in Josephus's account between the lunar eclipse and Passover for 13 Mar 4 BC (the Consensus View promoted by Emil Schurer) is discussed by Andrew Steinmann: his conclusion is that only 10 January 1 BC fits all the criteria. See page 12 and subsequent of Steinmann's article (linked at bottom of this post).
End of Second Ramble }}
See
"Josephus misdated the census of Quirinius" by John H. Rhoads (Journal of the Evangelical Theological Society, March 2011)
https://www.etsjets.org/files/JETS-PDFs/54/54-1/JETS_54-1_65-87_Rhoads.pdf
"When did Herod the Great reign?" by Andrew Steinmann, Novum Testamentum 51, 2009;
https://www.jstor.org/stable/25442624?read-now=1&refreqid=excelsior%3A0c06cf19ef6befdf56b9123d5754ba4a&seq=1#page_scan_tab_contents