Your son's father ALREADY HAS PARENTAL RIGHTS. And, yes, he can, at any time, petition the court for shared custody, visitation and such. Your asking for child support has nothing to do with custody and visitation.
If your son's father wants to see him, then work with him or the courts will work it out for you.
AND, remember, child support whether received or not is NOT the ticket to a father seeing his son, or not.
Child support has nothing to do with your SON's right to know his father.