No, the Tickets API won't return association data on its own.
Instead, you'll need to use the CRM Associations API. We will be using more of these "shared APIs" going forward.
For example, if you're using this endpoint and using a ticket ID as the objectId, you would use the definitionId16 to get associated contacts, 26 for companies, and 28 for deals, per the table in the overview documentation.
Is there a way to to include associations with a crm-objects query?
As @dclaysmith said. It is a shame that a minor detail like this costs 1000s of api calls which could be avoided if the associations could be included in the properties itself instead of as a separate endpoint and that too have to be called for each record.