Just about to give QuickTags a bit of a test drive. When I installed it, nothing worked.

I spotted the problem.

I don't have my wp-content folder in the usual place. I separated the wp-content from the core WordPress folder.

Your plugin is assuming it is in the default wp-content location. Easily fixed by changing:

$plugin_dir = basename(dirname(__FILE__));
$this->siteurl = $this->trailingslashit(get_option('siteurl'));
$this->js_path =  $this->siteurl . "wp-content/plugins/$plugin_dir/js/";
$this->css_path =  $this->siteurl . "wp-content/plugins/$plugin_dir/css/";
$this->usercss = get_bloginfo('stylesheet_directory').'/mdqtstyle.css';

with

$plugin_dir = plugin_dir_url( __FILE__ );
$this->siteurl = $this->trailingslashit(get_option('siteurl'));
$this->js_path =  $plugin_dir . "/js/";
$this->css_path =  $plugin_dir . "/css/";
$this->usercss = get_bloginfo('stylesheet_directory').'/mdqtstyle.css';