A woman cannot legally list a father on the birth certificate if the father is not there to sign an acknowledgement of paternity or a court has not issued an order establishing that a certain man is the legal father. I doubt that would have changed in the 24th century or you could have a bunch of women naming celebrities or royalty as the fathers of their children. I would assume no father is named on Jack Crusher’s official record of birth. Certainly, his father would not be listed as a man who had been dead for three decades or so when he was born. The only way that would have happened is if there had been DNA testing and an acknowledgement by Beverly that she had used the sperm of her dead husband, with his permission on the record, or embryos they had created before his death, to have a child.