The text in question is in Justin Martyr’s First Apology:

CHAPTER XXXIV — PLACE OF CHRIST’S BIRTH FORETOLD.

And hear what part of earth He was to be born in, as another prophet, Micah, foretold. He spoke thus: “And thou, Bethlehem, the land of Judah, art not the least among the princes of Judah; for out of thee shall come forth a Governor, who shall feed My people.” Now there is a village in the land of the Jews, thirty-five stadia from Jerusalem, in which Jesus Christ was born, as you can ascertain also from the registers of the taxing made under Cyrenius, your first procurator in Judaea.
This is false. And almost certainly not, this is another classic case of ancient Christian apologists inventing documentary evidence, or vaguely referring to archival evidence, to support their arguments that it would be practically impossible to follow up on. See also the letter of Marcus Aurelius about the Christians involved in the ‘rain miracle’ in Germany from Apollinaris, and Tertullian on Pilate’s letter to Tiberius confirming the divinity of Jesus. They’re all tendentious rhetorical ploys.

Legal records were deposited in archives in provincial capitals, but the Romans were not especially keen on preserving records, especially between administrative transitions, and the idea that a census that was a century and a half out of date would still be accessible is basically implausible.
The census of Quirinius took place in 6-7 CE, ten years after the death of Herod the Great. If Jesus was born during the first Roman census of Judea, he would have been 22 or 23 when he was crucified.

The census did not require anyone to travel to their ancestral cities. It did not apply to anyone living in Galilee. It would not have applied to Joseph at all. There is no reason it would have recorded birth records of children or even asked the names of anyone except men.

According to the Israeli Antiquities Authority, no confirmation has been found that Bethlehem was even occupied in the 1st Century. This is a quote from Aviram Oshri, Senior Archaeologist for the IAA in an article for Archaeology Magazine:
But while Luke and Matthew describe Bethlehem in Judea as the birthplace of Jesus, “Menorah,” the vast database of the Israel Antiquities Authority (IAA), describes Bethlehem as an “ancient site” with Iron Age material and the fourth-century Church of the Nativity and associated Byzantine and medieval buildings. But there is a complete absence of information for antiquities from the Herodian period–that is, from the time around the birth of Jesus.
Oshri thinks Jesus was born in a different town called Bethlehem that was in Galilee very close to Nazareth (which I think misses the point that Bethlehem was chosen by the evangelists specifically to meet expectations that Jesus would be born in the same town as David. That was the significance of Bethlehem, not the name), but the point is that Bethlehem may not even have been inhabited at the time of Jesus, but just an Iron Age ruin later revived and rebuilt by Byzantine Christians.

Josephus says that the Zealots burned down all the public archives to destroy debt records. The entire region was then systematically destroyed and burnt down, town by town, by Vespasian before Titus finished the job in Jerusalem.

The simplest answer is that Justin Martyr was repeating some random crazy rumor he heard. If you think about it, what would make a public record even identifiable as recording the birth of Jesus? Yeshua was a very common name. What would distinguish one baby Yeshua from another? They didn’t even have last names.