Depends what you mean by "Airport hotel" -- there is an airside hotel very close to the lounges, but I'm not aware that QR ever puts passengers there. The Oryx Rotana is a hotel about a ten-minute drive from the airport, and in my experience that's where QR puts passengers flying in F or J who qualify for the STPC program. Usually, that hotel also participates in the +Qatar voluntary stop-off program.

Note that DOH normally has five lounges:

• Al Safwa First Lounge
• Al Mourjan Business Lounge
• Qatar Airways First Class Lounge
• Qatar Airways Business Class Lounge
• Oryx Lounge

The Al Safwa is much nicer than the Al Mourjan, which is much nicer than the Qatar Airways First Class Lounge, which is nicer than the Qatar Airways Business Class Lounge. I'm not sure exactly where the Oryx Lounge fits, but it's not nicer than the Al Mourjan.

The Al Safwa is available to anyone flying in QR F (even regional F), and passengers can pay a (hefty) upcharge to access it instead of one of the other lounges. It has an order-off-the-menu dining room, and fully private rest suites with a bed, shower, toilet, and TV.

The Al Mourjan is available to anyone flying in QR J , and passengers can pay an upcharge to access it instead of one of the other lounges. It is quite large, with a second-floor buffet restaurant, multiple lounging areas downstairs, a sandwich restaurant, a family room, a game room, etc.

The Qatar Airways First Class Lounge is available to OneWorld Emeralds flying any OneWorld airline, but if an Emerald is flying QR J, the Al Mourjan is much nicer. The Qatar Airways Business Class Lounge is available to OneWorld Sapphires flying any OneWorld airline. The Oryx lounge access seems to be for Emeralds, Sapphires, and anyone flying OneWorld F or J.

See the QR lounge page, and the OneWorld DOH lounge page. The latter notes which of these lounges are currently closed due to COVID-19 shutdowns.