No system exist to register a firearm even if you wanted to.
The 2010 Florida Statutes(including Special Session A)790.335 Prohibition of registration of firearms; electronic records.—
(1) LEGISLATIVE FINDINGS AND INTENT.—
(a) The Legislature finds and declares that:
1. The right of individuals to keep and bear arms is guaranteed under both the Second Amendment to the United States Constitution and s. 8, Art. I of the State Constitution.
2. A list, record, or registry of legally owned firearms or law-abiding firearm owners is not a law enforcement tool and can become an instrument for profiling, harassing, or abusing law-abiding citizens based on their choice to own a firearm and exercise their Second Amendment right to keep and bear arms as guaranteed under the United States Constitution. Further, such a list, record, or registry has the potential to fall into the wrong hands and become a shopping list for thieves.
3. A list, record, or registry of legally owned firearms or law-abiding firearm owners is not a tool for fighting terrorism, but rather is an instrument that can be used as a means to profile innocent citizens and to harass and abuse American citizens based solely on their choice to own firearms and exercise their Second Amendment right to keep and bear arms as guaranteed under the United States Constitution.
4. Law-abiding firearm owners whose names have been illegally recorded in a list, record, or registry are entitled to redress.
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(2) PROHIBITIONS.—No state governmental agency or local government, special district, or other political subdivision or official, agent, or employee of such state or other governmental entity or any other person, public or private, shall knowingly and willfully keep or cause to be kept any list, record, or registry of privately owned firearms or any list, record, or registry of the owners of those firearms.
http://www.leg.state.fl.us/statutes/index.cfm?mode=View Statutes&SubMenu=1&App_mode=Display_Statute&Search_String=firearms&URL=0700-0799/0790
Note: there are exceptions which are covered at the link. edited for typo