I know that there is a Jira-to-Quality Center bridge that can be purchased, but have no experience with using it.
Our QA group uses Quality Center for writing and executing test cases, but since our development group uses Jira for bug tracking, QA also uses Jira.
The disadvantage, from a QA perspective, is that there is no automatic connection between a bug and the test cases that discovered it, users must manually paste the path to the test case. As can be expected, sometimes this gets left out of the ticket.
I have worked on projects where Quality Center was used by both dev and QA for bug tracking. I felt like it provided a better connection between the bug and the specific testing that discovered it.
I don't know what advantage Jira provides to the development side, but I've yet to meet a developer that likes QC.