Florida is a no fault state but not for property damage in a car accident. The Florida no-fault system really is in regards to medical payments. This means the insurance company covering your car would make payments for your bodily injury claims regardless of who was at fault in an accident.
The foundation of the FL no-fault law involves a trade-off of benefits: assuring payment of medical, disability (wage loss) and death benefits, regardless of fault, in return for a limitation on the right to sue for non-economic damages (pain and suffering).
For property damage to another vehicle in a rear end accident where you are at fault your insurance company should be covering the other party's damages to their vehicle. The property damage liability part of your auto insurance policy should cover this person's car where you damaged it in the accident.
The other person will likely contact their insurance company to inform them they were in an accident and especially if they were injured and needed to place a claim through their personal injury protection (PIP), which is also referred to as the no-fault portion of a car insurance policy.
Since you are the one found at fault for being unable to stop in time and rear ending the other car then you would be responsible for their property damage since in Florida no fault refers to medical issues, not property damage.