I still have my DVD case, and the serial number on that appears to be stored in the file:
~/Library/Preferences/com.aspyr.civ3conquests.plist
This is a binary format property list file, so you need to open it in something that can display it in human-readable form.
If you still have a copy of your old Users/<your user name> folder, open it, find the Library/Preferences folder within it, open com.aspyr.civ3conquests.plist in BBEdit or Xcode/Property List Editor, and look for the line:
<key>Serial</key>
<string> xxxxx-xxxxx-xxxxx-xxxxx-xxxxx</string>
The string value is your serial number.