I think this is intended to clamp onto an exhaust pipe. Home generators usually end in a muffler screen, not a pipe that you could clamp onto and I don't think this would work.

You need something like this

https://www.amazon.com/Champion-Inverter-Generator-Galvanized-Extension/dp/B07HBD8T9J/ref=sr_1_2?crid=3ADK3T4WHNTMT&keywords=generator+exhaust+extension+kit&qid=1640670932&sprefix=generator+ex%2Caps%2C81&sr=8-2

or the ability to weld a pipe onto your muffler.