Can a mom give up her rights to the dad through a notary?

A notary may be able to notarize the affidavit of relinquishment of parental righs, but that is the only thing a notary may do. Very specific language is required, and a notary is not qualified to advise you about the language in the affidavit.  This is a very technical area and most people require the representation of an attorney to do it correctly.  You will need to return to the court that made the orders in place now, and ask the court to termiante the rights of the mother.  Many courts will not agree to do that unless a step parent is simultaneousy petitioning the court to adopt the child, but I do not know what your court prefers in this regard.  The court makes the ultimate decision whether or not to terminate the mother's parental rights. IF the court does not make that order, then even if mom has signed an agreement, her rights are not termianted and her child support arrearage and obligation remain in place. The child support will not be automatically cancelled. The court order must state that the support obligation is terminated.  If the orders were done with the Office of Attorney General involved, most likely that office must also be involved in the suit and must approve whatever you are attempting to do with respect to the child support arrearage and child support obligation going forward.