If you are working with Green Spotted Puffers (GSPs), T. Nigroviridis, you should understand that they are a brackish to marine fish.. Although they can live as juveniles and short-term in freshwater, they need high amounts of salt to do well.
A black belly is a sign of these types of puffers being stressed. It should be pure white. If its black with white spotting like you say, it might be T. fluviatilis or sabahensis, but not likely as they are usually harder to come by.
There is a good chance your puffer has internal parasites, as they are very common in juvenile GSPs.
More than likely it is due to poor water quality, lack of tank size and simulation, and some other underlying stress or poor quality to begin with.
Do you add salt?
Do you have any ammonia, nitrite, or nitrates?
What size tank?
How many decorations?
What are you feeding?
You might check out **************.com, as they are experts in puffer husbandry and will be able to set you off on the right foot.. but if I were you I would do alot more reading next time before deciding you want to try GSPs again.. (I am speaking in past tense because this puffer will probably die, sorry, just being honest.)