I'm going to drop some head pounding, time sucking, information that has taken me the better part of 2 days to figure out, and every time I went searching this thread kept coming up so I feel this may be the best place to drop it for any future poor soul that tries to figure it out.
I have been trying to figure out why my 10th gen Intel Celeron Elkhart Lake J6413 would not show the QSV options. Even though in handbrakes activity screen it says "qsv is available on this system", I went down every rabbit hole, every random thread I could find. I followed all the things in this thread even though I'm running 6.11.5 and some of the things weren't needed.
I think the main issue I was running into was the fact that my CPU/GPU only supports low power mode (according to the jellyfin documentation). Which when I was running my server off of an i7-8700T, I never had a single issue, and the QSV options just came up.
When I was on the handbrake site looking at the QSV documentation,
https://handbrake.fr/docs/en/latest/technical/video-qsv.html
There was a link to a github "Known Issues and Limitations" for intel media drivers.
https://github.com/intel/media-driver#known-issues-and-limitations
This got me trying to put "options i915 enable_guc=2" in all the locations the "i915.conf" file may live.
What ended up working for me was putting "options i915 enable_guc=2" in the /boot/config/modprobe.d/i915.conf. This does copy down to the /etc/modprobe.d/i915.conf on reboot.
I hope this can help anyone in the future if you come upon this.