Is it true that there is no record of a census around the birth of Jesus and therefore Joseph and Mary weren’t in Bethlehem?

It is certainly true that some historians have doubts. Some historians have doubts about any historical event, and some are especially eager to disbelieve events recorded in Scripture. That doesn’t mean they have any evidence against what Scripture says. It is no surprise that there is no record of a census since there is no record of most events from this time. We don’t have Rome’s governmental archives. The historical records for things such as a census are spotty. (Of course, what your television show meant is “there is no record apart from Luke.” Luke is a record, but they are discounting it, despite the fact that Luke’s book of Acts is the work of a fine historian.) In short, the argument from silence proves nothing.