Can a person volutarily relinquish their parental rights in the state of Florida?

Terminating rights doesn't necessarily mean they are not responsible for paying child support
Checklist: Grounds for Terminating Parental Rights - Findlaw for the Public -

Signing Off From Parental Rights

Not without the other parents permission, and the permission of the court, normaly most courts will not alllow it, although there are provisions for it.

Also giving up your rights does not stop child support, you still have to pay child support even if you are allowed to give up your rigthts, rights and obligations are two separate issues
So if you don't want to use your rights, just don't use them, that is the same difference.