I have a four month old and me and the mom both want me on her brith papers, but we don't know how, can someone please help me?

You will have to be proven the father by DNA in most US states or be deemed the father by both statements from you and the mother.
But in all cases you will have to go though the court to get the name added.

There are two ways that a father's name can be added to the child's birth record when the parents are not married at the time of the child's birth and or the father for whatever reason was not present.

1) Recognition of Parentage (ROP) : The ROP form is also available through a county child support office in your governing county.

2) Court Ordered Paternity Adjudication : To request a court ordered paternity adjudication, contact the court administrator's office in the county where you live.

Tennessee Department of Health: Vital Records

Hope this helps you.