Why do you think it's GOG's fault and not your PC, especially if it works for so many people?

Do you have DirectX 9.0c (not 10 or over) manually installed (instructions here)?

Did you run the game as administrator, in compatibility mode for Windows XP SP3, and with fullscreen optimizations disabled?

Did you rummage through Google or, hell, this subforum for answers?

Got anything else? 'Cause none of that worked for me.

Setting seta net_reportUsageStatistics to "0" got rid of that annoying stats window but now the mouse just sticks itself in the top left corner.

Googling the issue either brings up issues with other games entirely, or points me back to the admin/compat settings/etc stuff or, if Steam, verify the game files (which still didn't fix anything for me anyways).

Funny part is that my mouse can happily leave the game and run onto my second monitor, almost as if the game isn't capturing it or something, and the only keyboard command it responds to is ALT-F4

Which graphics card are you using? AMD, for one, is known to break compatibility with older titles in driver updates.

At the moment, I'm trying to run it off my GTX 970, driver 411.70, Win 10 x64 (build 1809). Even went and re-ran the dxwebsetup to be sure the 1809 upgrade didn't break anything (it hasn't).

Yes I tried things you mentioned.
Nothing I worked

So I unplugged my Bluetooth adapter and suddenly things started working just fine.. Guess something was connecting over that as a controller/HID device that quake was picking up on

I guess try disabling bluetooth if you've got it