You can use a tracking link which redirects to the facebook page and allows you to know if the user clicked on the button. For instance : domain.tld/redirect.php?url=http://facebook.be...
If users are visitors, you can simply set a cookie and verify when you display the link if the cookie exists or not. But they can delete the cookie, so give us more details if security is important.
If users are logged, you can update a SQL row or something similar.
Another solution: set a cookie with javascript when the user clicks on the link, it's doesn't require to modify the link, just add an event