There is not a direct integration from TestComplete to TestRail but TestRail has a REST based API that you can easily send test cases to TestRail and collect test results.
Also both products integrate with JIRA which can be an intermediary
Please take a look at the REST API section on this page
/Alex [Community Hero]
[Community Heroes] are not employed by SmartBear Software but
are just volunteers who have some experience with the tools by SmartBear Software
and a desire to help others. Posts made by [Community Heroes]
may differ from the official policies of SmartBear Software and should be treated
as the own private opinion of their authors and under no circumstances as an
official answer from SmartBear Software.
The [Community Hero] signature is used with permission by SmartBear Software.