This is an example of a query which gets a single random event from the 'featured' event category amongsts evetns from The Events Calendar

page-random-event.php

<?php
/**
 * This is an example of a query which gets a single random event from the
 * 'featured' event category amongsts evetns from The Events Calendar
 *
 * @see http://tri.be/support/forums/topic/random-featured-event-custom-widget/
 */

get_header();

$featured_query = new WP_Query();
$featured_query->query(array(
	'post_type' => 'tribe_events',
	'posts_per_page' => '1',
	'orderby' => 'rand',
	'eventDisplay' => 'upcoming',
	'tax_query' => array(
		array(
			'taxonomy' => 'tribe_events_cat',
			'field' => 'slug',
			'terms' => 'featured',
			'operator' => 'IN'
		),
	),
));

if ($featured_query->have_posts()) :
	while ( $featured_query->have_posts() ) : $featured_query->the_post();
		// show the event here with whatever function you need
		die(var_dump($post));
	endwhile;
endif;

get_footer();