You can query the database from your module to get the Menu Description of your desired menu:
The code will look like:
$db = JFactory::getDbo();
$query = $db->getQuery( true );
$query->select( 'description' )
->from( '#__menu_types' )
->where( 'id = '{your menu item id, e.g. 1}' );
$db->setQuery( $query );
$description = $db->loadResult();