Call Ajax on Successful Facebook Share.

functions.php

<?php
add_action('wp_ajax_facebook_share', 'facebook_share', 0);
add_action('wp_ajax_nopriv_facebook_share', 'facebook_share');
function facebook_share() {
    $your_data = $_POST['your_data'];
    echo $your_data;
    die;
}

share.php

<a href="javascript:void(0)" class="fb-btn btn facebook-share" your-data="Custom-Date">Share</a>

<script src="http://connect.facebook.net/en_US/all.js" type="text/javascript" charset="utf-8"></script>
<div id="fb-root"></div>    
<script type="text/javascript" charset="utf-8">
FB.init({
    appId: 'Your-app-id', 
    status: true, 
    cookie: true,
    xfbml: true
});
/*****Facebook share call********/
jQuery('.facebook-share').on( 'click', function() {
	var yourData = jQuery(this).attr('your-data');
		FB.ui(
		   {
		     method: 'feed',
		     name: 'Share Name',
		     link: 'Share Link',
		     picture: 'http://fbrell.com/f8.jpg',
		     caption: 'Reference Documentation',
		     description: 'Share Content',
		     message: 'Facebook Dialogs are easy!'
		   },
		   function(response) {
		     if (response && response.post_id) {
	       		jQuery.ajax({
	       					type: 'POST',
	       					url: "<?php echo admin_url('admin-ajax.php'); ?>",
	       					data: {
	       						action: "facebook_share",
	       						your_data: yourData
	       					},
	       					success: function(result) {
	       						console.log(result);
	       					}
	       		});

		     } else {
		       alert('Post was not published.');
		     }
		   }
		 );
});
/******Facebook share end*****/