I want to sign my girlfriends daughters birth certificate and become her father. She already calls me dad and know no other father as he wants nothing to do with her. Can I do this with out getting in trouble?

No. You would be committing perjury. An acknowledgment of paternity is a legal document and swearing falsely is perjury. If the bio father challenges your paternity there could be trouble.

Marry your girlfriend and adopt the child.

No. That would be a fradulent act. Is the bio father not on the birth certificate?