Since Oculus Quest doesn't support SteamVR on Linux (AFAIK; they might have added support), not directly.

We'd have to implement a Godot VR interface for the Quest. If they have Linux support at all, this should be doable.

I have Rift not a Quest so I don't know if they're the same but you can pull your desktop and every window in your desktop up in VR on the default Rift OS, even in the middle of running another VR app. I had a browser open in the middle of playing Half Life: Alyx. Did the same on No Man's Sky VR

Someone asked if I could watch all the Star Wars movies at the same time. I got 6 trailers running at once in 6 separate virtual monitors with the default built in OS feature.

The Quest runs Android and is a stand-alone device, it's not just a display unit for your PC.

Yes, but there's the PC link option for the Quest, so if you have the link do you get the same experience as a Rift including being able to pull all your Windows desktop windows individually into VR?

Quest link requires the Oculus app which only runs on windows. ALVR/VirtualDesktop are also Windows only.

A cross-platform possibility would be if it served up a WebVR page that could be accessed with Firefox Reality.