no. after the cookie expires, the browser does not include the cookie with server requests.
if you want an event, then you will need to create one. on each request, you will need to log the timestamp. when timestamp + cookie timeout > now, then fire the event.
you could use a background service to do the monitoring and keep the list of users and their last access.
note: if hosted in a webform, then a database and a scheduled job might make sense,