[SemperFideles]

I think this should work (replace siteurl.com with yours, make sure you made a google.php file, and use your google code as opposed the the X's):

Replace

        <form action="search.php?do=process" method="post">
            <input type="hidden" name="s" value="{vb:raw session.sessionurl}" />
            <input type="hidden" name="securitytoken" value="{vb:raw bbuserinfo.securitytoken}" />
            <input type="hidden" name="do" value="process" />
            <span class="searchBox_container"><input type="text" value="" class="textbox" name="query" tabindex="99"/></span>
            <span class="searchBox_button"><input type="image" src="images/styles/BlueDemon/buttons/search.png" name="submit" onclick="document.getElementById('navbar_search').submit;" tabindex="100"/></span>
        </form>

With

<form action="http://www.siteurl.com/google.php" id="cse-search-box">
    <input type="hidden" name="cx" value="partner-pub-xxxxxxxxxxx:xxxxxx-xxxx" />
    <input type="hidden" name="cof" value="FORID:9" />
    <input type="hidden" name="ie" value="ISO-8859-1" />
             <span class="searchBox_container"><input type="text" value="" class="textbox" name="q" tabindex="99"/></span>
             <span class="searchBox_button"><input type="image" src="images/styles/BlueDemon/buttons/search.png" name="submit" onclick="document.getElementById('navbar_search').submit;" tabindex="100"/></span>
        </form>
<script type="text/javascript" src="http://www.google.com/cse/brand?form=cse-search-box&amp;lang=en"></script>