Out of the box? No
You can do some trickery to try and fool WP_Query into using your new db object but it may have issues with cached objects etc
e.g.
global $wpdb;
// backup wpdb
$thedb = $wpdb;
// pretend mydb is actually wpdb
$wpdb = $mydb;
// do stuff
// reset wpdb back
$wpdb = $thedb;
However if you’re trying to pull in data from another site, I would strongly recommend against remote SQL queries, if not for performance reasons but for security reasons.
Have you considered RSS aggregation plugins?