@znicholson both will likely work. Probably best to use the public API though. Do you have specific questions about how to use it? HubSpot writes their JavaScript docs using ES6 syntax which isn’t natively supported in all browsers and it can look a little odd. See below for a more supported JS syntax:
<script>window.HubSpotConversations.on('conversationStarted', function(payload){
dataLayer.push({'event': 'hs_chat_started'}); // This will need customized to your particular needs.
});</script>