Yes, if you are convicted of the New York speeding ticket the NYS courts will inform the Florida Department of Highway Safety and Motor Vehicles (HSMV) of this moving violation. The FL HSMV will then add it to your driving record and assess points.
If you hold a Florida license and have received a ticket in another state, they will send Florida the ticket information and it will be added to your record. You will receive points on your license if the ticket is a point-accessible violation according to Florida Statute 322.27(3). Florida law does not allow any school or program to remove points for a ticket received in another state. Any unpaid ticket reported by another state will result in the suspension of your Florida license. If this has happened you must contact the county where you received the ticket and obtain a receipt with the Court seal.
You can get a car insurance quote here to see how, if at all, this could affect your premiums.