Is it possible to configure DQIT to push notifications to the assignee(s) if a configured rule is triggered?
You can configure DQIT workflows for the path that issues take through DQIT. A number of configured actions can be created when transitions are executed, including sending of email notifications. The email notifications are configured as post actions in the transition section of Workflows. You can check out the documentation for it in DQIT Workflow Configuration page.