I don't think that there are any "out-of-the-box" solutions, but you can integrate FME quite easily with any cloud service that offers a REST API.
Zendesk has one for instance:https://developer.zendesk.com/rest_api
Basically all you need is the HTTPCaller and some JSON transformers and you're good to go.
Hi@juliarozema, I'm not familiar with a ticketing service system, but hopefully Email notification (publication) service provided by FME Server might help you.It's quite easy to configure/use Email publishers.See these links to learn more.