The directory is the template directory where you can find all template files. And you need to edit those.
But since you need to remove the avatar from random quote module this is not the place for it.
Remove below code
modules/mod_randomquotes/tmpl/default.php
<div class="quote-avatar">
<?php
if(JFile::exists(JPATH_ROOT.DS.'media'.DS.'communityquotes'.DS.'images'.DS.$quote->avatar)){
$avatar = $q_root_path_uri.'/media/communityquotes/images/'.$quote->avatar;
}else{
$avatar = $q_root_path_uri.'/components/com_communityquotes/assets/images/default_avatar.png';
}
$avatar = JHTML::image($avatar, $quote->author_name, 'height="48px" width="48px"');
echo JHTML::link(JRoute::_('index.php?option=com_communityquotes&view=quotes&task=aprofile&id='.$quote->author.':'.$quote->author_alias.$rq_itemid), $avatar);
?>
</div>