Hi, @Pier.
Since this discussion was last active, we released a Conversations Live Chat Widget API. The features include these widget events, which you could use to trigger a Google Tag Manager event.
For example, the following script would listen for the start of a chat and trigger a GTM event named hs_chat_started.
window.HubSpotConversations.on('conversationStarted', payload => {
dataLayer.push({'event': 'hs_chat_started'});
});