Unfortunately at this time there is no direct integration. A workaround would be to use the XRay Jira REST API and create a parser for the JUnit style XML results file generated by TestComplete through the /exportsummary command line argument or the log.saveresultas function.
Here is an Atlassian Community post containing some details as well.
Solutions Engineer @ SmartBear
Hi @nucida -
I also would like to direct you towards this post which talks about how a REST API was used to create connection - it shows some great examples of the code.
Let me know if you have further questions.