Usually (as in nearly always) the generator does not have a dedicated battery; it typically cranks from either the chassis or house battery. On my '98 Coachmen, it's the house battery.
The house battery charges from the converter and from the engine when it's running, provided everything is working properly. The chassis battery charges from the engine, of course. Sometimes the chassis battery is charged from the converter via a bidirectional isolator or some other device, but that's comparatively uncommon on class C's I believe. If you want it to charge that way, you can install a Trik-L-Start (which is not a super expensive gizmo); it sends a few amps to the chassis battery when the converter is operating or something else (such as solar) is charging the house battery.
The generator charges the same things that charge from shore power, through the same mechanisms.
Have you verified that there actually is a battery in the battery box by the generator, and it's not just an enclosure for some other electrical equipment?
If you have a voltmeter, it's easy to check what batteries are charging under different conditions. If the voltage is above 13V, usually 13.5V or higher, it's charging or was just charged and still has a surface charge. If it's less than 13V, usually 12.6V or less, it's not being charged.
