as of today, we cannot trigger testcomplete tests from within zephyr for jira. the closest functionality would be the Test Automation page (formerly called ATOM beta), which would essentially gather the results from previously run tests.
In addition to @hkim5s response, which is correct, the integration between Zephyr for Jira and TestComplete is native to TestComplete. You can link a TestComplete test to a Zephyr for Jira test execution so that wherever it is being executed (local machine, Jenkins etc.) the results will feed into Jira automatically.