It is very easy to track live chat event by google analytics. For example, Your website has zendesk live chat then insert below code in zendesk api code line (below zendesk api code),

Zendesk.setCallback('zendesk', function('email', 'msg', 'type'){
if (typeof(ga) == 'function') {
 ga('send', 'event', 'Zendesk', 'chat', email);
}else if (typeof(_gaTracker) == 'function') {
 _getTracker('send', 'event', 'Zendesk', 'chat', email);
} else {
 _gaq.push(['_trackEvent', 'Zendesk', 'chat', email]);

After that code in your website, You can see detail in google analytics in event submenu in behavior report.