The link is to a site that takes up the argument (mentioned in the Staff Report) that Quirinius served twice, and so the census mentioned was taken under his earlier rule, a decade earlier.
That argument has no supporting evidence… I think the link (I looked at it only quickly) provided by JThunder says there an inscription that possibly mentions that someone held a some office at two different times, and that could possibly be Quirinius.
The article makes one other point that is just plain stupid. It argues that there WAS a census of all the Roman Empire because Augustus says that he was made emperor “by the authority of the senate and all the Roman people” (I may have got the quote slightly wrong.) The article says that he could not have been elected emperor “by all the Roman people” without some sort of empire-wide vote, and that’s just silly. Emperors ALWAYS say they were elected by “the people” even if there were no election; and the phrase “Senate and People of Rome” (SPQR) was used commonly to maintain the illusion of democracy.
None of those arguments, even if you grant them, can overcome the objection that no authority in power would conceivable order a census that required people to travel to the home city of a remote ancestor. Such a requirement would have disrupted trade, lives, and – the PURPOSE of such a census – tax-collection.
In short, while there may (conceivably) be an argument that has a census connected with Jesus’ birth, there is no argument that such a census would require Joseph to move to Bethlehem where his many-many-generations ago grandfather lived.