Short answer is no.
Only the menu and components can define sef routes. The other option is to run something early in the life cycle to mess with the routing directly, but the template runs much too late for that. A system plugin would be the extension for this.
This template allows you to output as blank:
If you look at the tmpl_mode option for clues.
Also - this code could perhaps work in identifying the component (com_ajax) in your index.php:
//ECHO COMPONENT TYPE ()
$jinput = JFactory::getApplication()->input;
if(($jinput->get('option') == '...