You'll need the exact model of your generac, and then, with the manuals for that model, it will explain how the charging of the battery happens. There's no alternator, but there is a charging module.
On my 22kw model, it doesn't charge the battery while running. Installation guide says run a separate 120v power line back out to the generac, and that is used to supply the charging module, which charges the battery (when the unit is not running; yes ... nobody can believe this is how it works).
Everyone hates this concept, wrt generacs, so I defeated it by putting a smart 12v battery charger on the battery, fed from an extension cord. Whenever it sensed that the battery wasn't optimal, it trickle-charged the battery.
You can charge the battery back up with any 12v battery charger; get a trickle-charger on it asap. If you think the battery is charged, but genny won't start, use a load tester on the battery.
Hope this helps ...