I don’t see much to refute. Aside from Luke’s Gospel, there is no historical record of this census. This leads to three possibilities. One, the census never took place. Two, the census did take place, but Luke got some of the details wrong. Three, the census took place exactly as Luke relates.
Interestingly enough, the linked article doesn’t claim there wasn’t a census. In fact, it makes the opposite claim. For example: “For a variety of political reasons, the Emperor Augustus created a new province called Judaea, around the year 6 AD. At that time, Syria was governed by a legate, a person of senatorial rank, named P. Sulpicius Quirinius; the new province would be governed by a procurator (of lower, equestrian rank). It was common practice to take a census upon the creation of a new province, and so Quirinius ordered a census to be taken for Judaea.”
The sticking point is not that the census never happened, but the timing of the census in relation to Jesus’ birth year: “P. Sculpinius Quirinius was legate (governor) of Syria in the years 6 - 7 AD. He did order a census. However, the assumption that Jesus was born in the year of Quirinius’s census (6 AD) leads to irreconcilable chronological problems in the subsequent events of his life. It is entirely unlikely that Jesus was born in the year of Quirinius’s census; most scholars put Jesus’ birth around 4 BC, a good ten years before Quirinius’s census.”
Ah, but there is more to the story. Tacitus (Annals 3:48) and Florus (Roman History 2:31) speak of Quirinius leading military expeditions in the eastern provinces of the Roman empire a decade before his governorship of Syria. Various forms of “joint rule” were common in the ancient world, so it is possible that Quirinius’s military role included some sort of governorship. Two fragmentary Latin inscriptions give some credence to an earlier term of office for Quirinius.
It is thus possible that in his earlier term of office, Quirinius called for a local census around the time of Jesus’ birth year. Then, later, there was a general census when Quirinius became legate circa A.D. 6. Luke simply conflates the two incidents, hardly an earth-shattering admission.