Did saving continue to work once you made enough quicksaves to start overwriting the oldest ones (there are 5 kept by default, which can be increased up to 25 in the options)?
If you were able to overwrite a named save (if you tested that), it shouldn't be an issue with quicksaves.
Some other suggestions not mentioned above, or expanded upon (though they didn't help carolus):
Try creating a new profile, and then start a new game to test saving / quicksaving.
Shut down all non-essential programs before starting the game, especially anti-virus and any backup programs that may be monitoring the My Documents folder.
The crashing when quicksaving can be caused by backup or sync programs, if they lock the original save.
A few people have had BitDefender block the game from being able to create files and folders, which has can also cause similar problems. In that case, including both the game executable and Larian Studios My Documents folder in BitDefender's trusted app and directory sections should fix the problem (Advance Threat Defense, Ransomware Protection / Safe Files, Application Access).
Try browsing to the '..\Documents\Larian Studios\Divinity Original Sin 2\LevelCache' folder and delete the contents. A corrupt file there can cause problems when saving or loading.
Try creating a new Windows administrator user account (either with the My Documents in the default location, or on a different drive, if available), switch to that account and try starting the game from there, directly from the executable.
Try doing a clean boot and then test the game. Click Start, or hit WinKey-R, type in msconfig and hit enter; in the General tab, click Selective Startup, uncheck Load startup items (if required) and leave Load system services and Use original boot configuration options checked. Next, click on the Services tab, check the box to Hide all Microsoft services, then click the Disable All button (maybe make a note of which are currently enabled/disabled), then click OK and reboot the computer.
Run msconfig again to switch back to the normal boot configuration.
