I'm playing Hollow Knight on Windows 11 and without GOG Galaxy.
The game doesn't just not unlock achievements, it doesn't show achievements at all, when looking them up in the menu.
It's just an empty list and the scrollbar can be moved until it leaves the screen.

I only own the Steam version, but here the behaviour is just the same (when running the game without Steam): it works just fine, except the internal "achievement browser" doesn't show anything.

I found this solution: in the subdirectory hollow_knight_Data\Plugins\x86_64\ there's a file named steam_api64.dll (*) - delete it and the internal achievements start working

It's possible the GOG version still has this file (even if it's unused); or possibly there's something like galaxy_api.dll in the GOG version - try deleting both. Good luck! :-)

((*) In Linux, it's hollow_knight_Data/Plugins/libsteam_api.so.)