It sounds like you need a `meta_query`.

    $query = new WP_Query ( 
      array( 
        'paged' => $paged, 
        'post_type' => 'event', 
        'posts_per_page' => 4,
        'meta_query' => array(
          array(
            'key' => 'date_key_name',
            'value' => date('Y-m-d'),
            'compare' => '>=',
          )
        )
      )
    );

Be aware that your dates need to be in some format that [a machine can parse correctly][1].

##Reference
http://codex.wordpress.org/Class_Reference/WP_Query#Custom_Field_Parameters


  [1]: https://wordpress.stackexchange.com/a/75318/21376