I am going to try and answer your question, using info I found on the internet, since I don't do this kind of billing. Hopefully someone here who does do this kind of billing will jump in also, and verify if what I found is correct.
Once the baby is born, they are considered an individual, and as such, need to either be added to the mother or father's insurance, OR they need to be insured under their own policy. My understanding is you cannot bill services rendered to the baby under the mother's info, as though she was the patient, because the baby is now an individual.