add filter to “quick edit menu” in wordpress admin

You do not want to reference the global $post, but the post that is given to you as one of the argument. You simply need to remove global $post;

Remember also to sanitize input and prefix function names.

function wpse50651_filter_transition_post_status( $new_status, $old_status, $post ) { 

    global $wpdb;

    $wpdb->query( 
        $wpdb->prepare( 
          "UPDATE my_table SET post_status=%s WHERE post_id=%d", 
           $new_status,$post->ID
         ) 
     );
}
add_action('transition_post_status', 'wpse50651_filter_transition_post_status', 10, 3);