Not likely. Generally, if the generator starts, the battery has done it's job.
My coach is a 2001 Alpine. I made an illustration that shows how things are powered in my coach. My generator starts off the house batteries. Power from the generator is routed through the transfer switch, then to the main AC panel. The inverter is powered from the main AC panel and charges the batteries.
Some coaches are made where the generator/shore power charges the house batteries, and the alternator charges the chassis batteries. Many coaches have equipment that allow cross charging from either source. On my coach, it is the echo charger that performs this function.
You haven't told us much about what your problem is except that your generator shuts down. How long does it run? How often does this happen? Will it restart? Have you tested your batteries? Do you know which battery bank starts your generator? I can make some wild guesses, but they are just that - guesses. I'd first suspect fuel level/fuel filter. Then I would be concerned about oil level. Then I would look for loose connections.