I saw this guide from the forum, however I believe it was somehow not correct, and missed some information (well, the post cannot be replied so I post some amendment here).
There are 2 packages: intel-media-va-driver & intel-media-va-driver-non-free, the "non-free" one is of course in "non-free" repo, it has all functionalities of original one (decoding) + proprietary (encoding) function, which is where the Quick Sync encoding acceleration coming from. I have compared both and the proprietary Quick Sync is much faster than VAAPI implementation. Tested with Jellyfin website's 120Mbps 4K HEVC 10bit SDR video, transcoding to 80Mbps 4K H.264, with VAAPI I get 15-20 fps but with Quick sync I can get 40fps+, which is a huge difference.
Original OP's setup will work with "VAAPI acceleration" setting under Jellyfin, which you still get hardware acceleration but it's open sourced implementation.
If you need the real Intel Quick Sync you have to install "intel-media-va-driver-non-free", but not from stable repo, that version is very very old, to get it work I have to install the version from "bullseye" (which is v21.1 now).
Also please note that, your Intel CPU version matters, the "intel-media-va-driver"/"intel-media-va-driver-non-free" are for CPU 9th Gen or above (e.g. Coffee Lake, Comet Lake, Gemini Lake), 9th Gen or older generation needs to use "i965-va-driver"/"i965-va-driver-shaders" (later one is with proprietary encoding driver). Interesting fact here is 9th Gen overlapping between these 2 drivers, and yes both of them will work, but "i965" one has less features (e.g. HEVC encoding) and slower performance when compared with the newer driver. I owned a Celeron J4125 which is a Gemini Lake Refresh, confirmed both driver were working but of course I would go for a new one.
Last note is, depending on which container you are using, if your LXC is Debian then you should be fine with Jellyfin 10.7.5, with Ubuntu 20.04 LXC you'll find that Quick Sync won't work even with new driver, it's due to older FFMPEG version (In Ubuntu Jellyfin bundled 4.3.1-4 while Debian has 4.3.2-1), manually updating to 4.3.2-1 will fix the problem.