As far as I understand bible "history" and standard history and the records of population available from the time there is no record of jc existence
Given the time frame I would be surprised even some records still exist
Thought bubble - who would Mary put as the father of jc? She has two options, god or Joseph
If the birth record was found and dad was recorded as god case closed. If Joseph I doubt they would be the only married couple with the names of Joseph and Mary
And I suspect given the movement of the birthdate of jc, around the calendar, until the church decided to steal a very active pagan celebratory day, 25th December, even the year would be suspect