There are plenty answers on WPSE about restricting page views based on logged in users or visitors.
what you should do is first check if user is logged in then check if you are on your restricted category and redirect on another page (maybe home or login page) if user is not logged in.
something like this in your functions.php file.
add_action( 'template_redirect', 'wpse_restrict_support');
function wpse_restrict_support(){
if( ! is_user_logged_in() && is_category( 'support' ) ) {
wp_redirect( '/wp-login.php' );
exit;
}
}
check out is_category(), is_user_logged_in() and wp_redirect() for more info