You could probably do that with our trigger plugin:
https://moodle.org/plugins/tool_trigger
Create a listener on the course complete event and the add an action to trigger an external web services call. I'm not sure if Jira has a webservice that will do exactly what you need but you might be able to send an action from tool-trigger to ifttt.com and then get ifttt.com to do what you want in jira?