I use the "classic" installers, so I don't know if it's any different with Galaxy.
The desktop icon is usually a shortcut that links to another executable file (with the original game icon). You can check what file it is linking to, find it in the installation folder and create a new shortcut specifically for the executable file, which will have the icon you want.
For example, The Witcher 1 launcher is a shortcut called "Launch The Witcher Enhanced Edition Director's Cut", which links to the executable file "launcher.exe" (to know where it is linking, right-click the shortcut and select "Properties"). Find launcher.exe, right-click it and create a new shortcut, et voilà. This is similar for the rest of GOG games. I wasn't able to find the original icon for all the games I have installed, though (most notable those using ScummVM - maybe they are set up in another way).