There are a few hotels in Bermuda near the airport, with the closest being Grotto Bay just 3 minutes across the bridge (causeway) from the airport.
Rosewood Tucker’s Point & The Loren offer luxury options that are nearby. While Aunt Nea’s Inn offers an affordable option.
However, Bermuda is only 21 square miles, and so if you are OK with traveling 20-45 minutes away from the airport, you can get to every hotel on the island.