I think it would be wiser to simply put it away and avoid starting the main engine over the course of the winter. Running it without getting it up to operating temperature will do more harm than letting it sit. Driving around for an hour in the winter is arguably better for the engine (though how much better is quite up to discussion), but very likely to expose the chassis and underside to road salt etc. that I for one would prefer to avoid.
Starting the generator to exercise it under load for fifteen minutes or so a few times is a good idea--not so much for the generator engine as for the brushes and commutator.
My riding lawn mower, for example, has sat unstarted every winter for something like twenty years, and it still works fine. It is starting to wear out, but that's not from sitting. It, incidentally, happens to have an engine made by Onan.