In Firefox, the search history that appears below the search box when you type in the search box is stored by your browser, not by Google Desktop.
To disable this feature in Mozilla Firefox 2.0:
1. Go to "Firefox" > "Preferences..."
2. Select "Privacy" > "Clear Now..."
3. Select the "Browsing History" checkbox.
4. Click "Clear Private Data Now."
5. Deselect "Remember what I enter in forms and the search bar."
6. Close the Preferences window.