To find this information you'll need some implementation. You'll want to have two custom events fire: one when a user is presented with the chat button, and another when the user clicks the button.
You'll be able to see volume and clickthrough, but also on which pages people see/click on the button.
For the first event, I recommend a page load rule that looks for a specific html element unique to the chat button, such as a div or section with a specific class. When it sees your chat button it fires
s.linkTrackVars = 'events';
s.linkTrackEvents = 'event1';
s.events = 'event1';
For the second event, I recommend a event based rule that listens for clicks on that same specific html element unique to the chat button. It should be an s.tl(); rule so that it does not increment a pageview. You'll probably also want to delay link activation. When it sees the click it fires
s.linkTrackVars = 'events';
s.linkTrackEvents = 'event2';
s.events = 'event2';
The event numbers above assume you're not currently using those events to track something else. You would do the above in DTM, with the code in the custom code of the Adobe Analytics section. Then in Adobe Analytics you'd configure the event.
Reference:
Success Events
Configure success events
What is a Custom Event?
