This is, more than anything, a Note to Self for the next time I do a clean install of OpenSUSE and can't figure out how to make Firefox stop making incredibly loud error beeps every single time I type Ctrl-F and type a search term that it can't find.
Disabling audio notifications in KDE's Configure Desktop has no effect on Firefox, because it uses GNOME's audio event system, not KDE's. (Not sure why it also completely ignores current system volume and always blasts the error beep so friggin' loudly; I'm sure there's a configuration setting for that somewhere.)
The program for controlling GNOME/GTK audio events is called pavucontrol (for Pulse Audio Volume Control).
Run that, go to the Playback tab, and click the Mute icon next to System Sounds.
This is really the sort of thing OpenSUSE should fix, seeing as its default configuration is a KDE desktop with Firefox as the default browser. It would be nice for YaST to have some kind of integrated audio notifications configuration tool, or, at the very list, to document pavucontrol better.