Currently, HubSpot does not display in the Help Desk ticket or chat conversation thread when a CSAT survey is sent. While you can view how many people have been surveyed (and their responses) in your survey tool, there is no note or visible event inside the individual ticket or chat conversation itself indicating that the survey was sent.
It would be beneficial to see that CSAT survey has been sent in the Help Desk Tticket or chat conversation thread.