After enabling the Agent Workspace, to obtain a report with the ticket ID associated with each chat, you need to export the file from the Zendesk Support instance. For details on how to request this information, see this article: Exporting data to a JSON, CSV, or XML file.
In the example below, you can see how the CSV option looks. The report lists the tickets from all channels. Right-click any of the Chat cells and select Filter Table by "Chat" to see only chat tickets.
The ticket ID information is displayed in the Id column, which is the second column from the left.