In the U.S., birth records weren't "officially" kept by governmental agencies until more recent times so a "certificate" of birth would probably not exist for some earlier generations of family members.
There are records, however, of what were called "bastardy" records for situations in which children were born out of wedlock. If you've heard of "bastardy bonds", the phrase applies to situations involving a child born out of wedlock. These court actions are similar to modern court filings to establish paternity except there weren't limitations as to people eligible to file such actions or people eligible to receive information about the details of the case.
If the child was born in earlier time periods, I'd check local resources for these types of records.