you could, and it shouldn't cause a problem on a gasser, but it could.
really, your not goign to gain much more than just routing the exhaust down under the truck.
think about all the air compressor threads where building an insulated closet is about the same as muffling the intake. the exhaust gets blame for the noise, but its a small part. and going into the trucks system isn't worth the effort.
if you want quiet, you need to invest in another generator. or proper sound deadining. fwiw, i dont think the modern weld/gens are worth the price if my 325efi that is supposed to be one of the quiet ones, goes down... i'm probably going back to running a Commercial Mobile Quiet Deisel. the harbor freight invertor gens amazing for what they are too, and they can be paralleled for more power.
a pto gen is going to be the best quiet option.
the most quiet setup i have ever had was the tb325efi inside of an e350. back doors, and at least one front window or door was alwyas open. made good airflow and never smelled of fuel or exhaust. the exhaust was stock and just went out the back. i did one temp exhaust re route, but it didn't help any.