I have a desktop system with HDMI graphics on the motherboard and a
graphics card. Here's the output from inxi -G:
Device-1: Intel Raptor Lake-S GT1 [UHD Graphics 770] driver: i915 v:
kernel
Device-2: NVIDIA GM107GL [Quadro K2200] driver: nouveau v: kernel
It's running Debian 12.11 with kernel 6.1.140-1.
If I connect the HDMI monitor before boot, it works. If I connect the
HDMI monitor after the system is running, it doesn't work.
(Applications -> Settings -> Display). I expect other desktops have
something similar.