The default folder is not set to My Documents (though the save games go there under My Games\GalCiv2).
Per the above comments, Stardock Central has an option to prompt for the installation folder.
If you've already installed it, you don't have to set the option and redownload. As the bits above say, use Regedit.exe and adjust the Path value at HKEY_LOCAL_MACHINE\SOFTWARE\Stardock\Drengin.net\GalCiv2 to point to the location you move to. If you're not comfortable with regedit, just try not to, I don't know, hit the Delete key? 
Kris