A company I work with recently looked at the same concept. We never made it further than the "can this happen" phase, but it appears that all the hard work will have to be done on the JIRA side. Quality Center provides a COM-based API that can be used to access data in Quality Center. Quality Center does run client-side code for each user which will allow you to respond to certain events, but we didn't like putting any of the JIRA integration there.
So using the COM API, you can make it happen, but you'll have to create the bridge yourself. Hope it works out.
There is another company that is working on an intergration between Jira and TD - Go2Group. Refer to this part of their website http://www.go2group.com/Go2Group-Sol...gin-Basic.html
Their Premium product is not yet available. I contacted them and got on a mailing list for more information.
(Opinions and information contained in this post are wholly my own and do not reflect the opinions of my employer.)