Using AJAX in wordpress theme

You need to add the $where parameter to your update statement because it doesn’t know which row to update. The $wpdb->update() method has no default value for the $where argument so you must supply one.

http://codex.wordpress.org/Class_Reference/wpdb#UPDATE_rows

Try this:

$wpdb->update( $wpdb->wp_frm_items, 
        array( 'alerts' =>  1),
        array( 'user' => get_current_user_id() )
        );