Family Law Questions? Ask a Family Lawyer Online.
Yes, you can file a petition to terminate parental rights with the family law court of the child's county of residence. If the father will sign a waiver of parental rights, you can file it with the court, and getting an order is simply a formality.
Giving up his parental rights does not take away his obligation to pay child support, though.
You should see a family law attorney in your area about getting this done.