What @Sam-Holder said is good; adding to that because I'm familiar with the issue and the tools you're talking about.
You're probably used to the idea that Quality Centre contains a bunch of test scripts, some of which pass and some of which don't (yet).
When you're doing BDD with automated scenarios, they pretty much always pass, all the time. Half the things that QC does simply aren't needed with modern Agile processes.
A pretty common practice is to put the scenarios into the Jira story until they're automated. They're transient. Nobody ever looks at Jira once the story's done. The codebase is the single repository of truth, and anything that lives in Jira gets ignored.
The automated scenarios are checked into the same codebase as the code. If the scenarios go red (fail), the team makes them green ASAP. They provide living documentation for the code. See if you can find someone to show you what Jenkins looks like in action and you'll get a better picture. Commonly the Jira number is added to the check-in comment to provide some level of traceability.
I think it's good practice to keep any manual test cases checked in alongside the automated ones (though please do question why they're not automated; if you automated them in QC you can usually do it with SpecFlow). This helps the test cases (scenarios) to provide living documentation for the code. In fact, getting rid of the word "test" was part of the reason why BDD came about, because BDD's really about analysis and exploration through conversation. It provides tests as a nice by-product.
To answer the question, the most commonly used tool at the moment is Git (at least for newcomers). It's version control that the devs are using. SVN / Mercurial are also OK flavours of version control. Get the devs to help you.
If you're still working in a silo and not talking to the devs, don't use BDD tools like SpecFlow - you'll find them harder to keep up-to-date, because your steps will probably be too detailed and English is trickier to refactor than code.
Better still, use the BDD tools and go talk to the devs and the BAs / SMEs / Product Owner who understands the problem. Get them to help you write the scenarios. When you start having the conversations, even over legacy code, you'll start understanding why BDD works.
Here's a blog post I wrote on using BDD with legacy systems that might also be helpful to you. Here's a blog post on the BDD lifecycle: exploration, specification and test by example. And here's one to get you started on how to derive that "Given, When, Then" syntax using real conversations.
