I looked up this topic on this sub, but there are very few answers. Can someone knowledgeable in the Biblical field help me out here?
Yes, the census happened. But that doesn't mean the story fits exactly.
In Luke, it says that this census happened while Quirrinus was governor of Syria: "In those days a decree went out from Emperor Augustus that all the world should be registered. This was the first registration and was taken while Quirinius was governor of Syria. All went to their own towns to be registered. Joseph also went from the town of Nazareth in Galilee to Judea, to the city of David called Bethlehem, because he was descended from the house and family of David. He went to be registered with Mary, to whom he was engaged and who was expecting a child. (Luke 2:1-5)"
In 6 AD Judea became a Roman province, as it's ruler, the Ethnarch Herod, was so unpopular. Quirrinus held a census of the new province for tax purposes. But there was no census in Galilee (which was not part of a Roman province: it was under a client ruler, the Tetrarch Herod). Also, Roman censuses did not require people to travel to their home town. Then, we've got the issue that Luke 1 says "In the days of King Herod..." But, there was no King Herod when the census was taken, he'd been dead for 10 years. Perhaps Luke is confusing the titles of Tetrarch and King? Or even the persons of King Herod and Tetrarch (or Ethnarch) Herod? Herod the Great had three sons called Herod, all of whom became rulers, and some of whom had sons called Herod, and frankly, it is pretty confusing! But that still doesn't really explain why a guy from Nazareth (no census) has to return to his hometown, as that's not how Roman censuses worked. Maybe... he lived there generally and was just in Nazareth for a short time? But then, why did he go and live in Nazareth for the rest of his life after the census? As you can see, it's not an utterly ridiculous story; it does link to real events, but it doesn't quite fit together.
It's probably worth noting that Matthew's gospel mentions nothing about a census. In Matthew's version, Jesus is born in Bethlehem (presumably because Joseph and Mary just live there, it's not explained), during the reign of King Herod (who, as mentioned, died 10 years before this census), but they flee to Egypt. When Herod dies: "But when he heard that [Herod] Archelaus was reigning in Judea in place of his father Herod, he was afraid to go there. Having been warned in a dream, he withdrew to the district of Galilee, and he went and lived in a town called Nazareth." (Matt 2:22-23)" Of course, Nazareth was ruled by a different Herod (Antipas), but as mentioned, Herod Archelaus was very unpopular (shortly after his father's death, he sent soldiers into the Temple to put down a protest, killing 3000 protestors).
As you can see, the gospels agree that Jesus was a)born in Bethlehem, and b)grew up in Nazareth. However, they don't agree on the details.
r/AcademicBiblical would a great place to ask this question as well.
While this answer by u/talondearg doesn't exactly address your question regarding the census it does answer what I suspect might be your underlying question of if Jesus was an actual person in history.