Not good to vent exhaust under the eave.
If any distance below the eve, sometimes rain will angle in and hit it.
More important, hot exhaust rises and can collected under the eve and enter the home. Carbon monoxide is deadly and people have been killed by it from a generator outside.
This link suggests permanently installed generators be 18" away from house, but portable generators be 20' away.
Perhaps the difference is due to members of the public locating it near openings vs. a professional installer mounting in a suitable location.
While a whole house generator provides many advantages during emergencies, it can also pose harm if not placed at a safe distance. Generator exhaust contains…
pickgenerators.com
Gas appliances use double-wall exhaust pipe and end a distance above the roof.
In the case of water heater, it has an opening just above water heater, Venturi effect to draw gasses that are around the water heater.