You have two possible approaches. Any administrative or legal action will need to take place in Florida.
You can look at:
That website will help you understand that your grandson will either need to (a) get an administrative order from the Florida Department of Revenue, or (b) get a court order from the District Court where the child lives.
The least expensive approach will be to ask for the Florida Department of Revenue to establish your grandson's paternity through the administrative approach.
If other issues can be worked out between your grandson and the mother, again, this would be the least expensive approach.
If the mother is going to continue to resist any visitation
with the child by your grandson, then he may have to get a court order. See, http://www.myfloridalaw.com/child-custody-law/establishing-paternity-in-florida/
will guide you to the closest office for the Florida Department of Revenue.