There is the OTRS family of ticket trackers. I've only used them as a customer, but I think you can do at least requirement tracking with them as well. I'm not sure whether they have a workflow for tests (test definition and executions of the test definitions) though.

1. OTRS Pro (closed source)
2. OTRS community edition
3. OTOBO - fork of the last open version of OTRS, professional support
4. Znuny - another fork, also professional support

In the medium run, Jira has an OpenAPI spec so you it is relatively easy to synchronize/import data into Jira if you can get your data into the representation of the Jira schema.