I was looking at the FL state laws for having a gun in a "private conveyance" usually meaning a car. As most know, you can have a loaded gun inside a "private conveyance", but it has to be "securely encased", i.e. in the glovebox or a zippered bag or something like that. My question is, does this apply to bicycles and motorcycles, which don't exactly have an interior? Would it be legal to have the gun in a bag or box attached to the bike? Under the seat of a motorcycle?

Are they? Yes.

Would it be legal? No, not without a carry permit.

That would depend on how your particular state defines "conveyance".  In most states, a bicycle is a vehicle, subject to the same traffice rules and regulations which affect a car or truck - with the exception of those specifically applied only to motor vehicles.  A bicycle is not a motor vehicle.

If Florida defines a bicycle as a vehicle (although not a motor vehicle), then you can transport a securely encased firearm on your bicycle without violating the law.  If so, get a locking saddlebag, and you'd be good to go.

I know a guy who got a DUI for riding his bike drunk. It seems Ohio considers a bike a bonafide vehicle if you are riding it on a public roadway.

Same here in NY. Only ever ticketed one, after he caused an accident. Boy, was HE mad!