The Concierge 1.3.2 Event Broadcasting Guide describes how to listen for the surveyCompleted event, which is intended to indicate when a website visitor completes a post-chat survey from the Concierge Visitor Client.
A recent issue was discovered wherein the surveyCompleted event is broadcast not only when the visitor successfully submits the survey form, but also if the visitor clicks the "No Thanks" button on the form itself. This can lead to false-positive events, and therefore should not be used for counting the number of successfully completed surveys. This will be updated in the documentation for the next release of the Concierge application.
The number of successfully completed surveys is still available from a Channels report titled "Survey Summary Data: # of Times Survey Submitted".