If you’re working at a company in QA and you have an existing business development team, I’d suggest talking to your manager about taking on some new projects with the other team. If you’ve been performing well, your manager should want to keep you happy and help you learn new skills. From the perspective of the existing business development team, it may take awhile to get onboarded but I’m sure they would be happy to get some extra help. If you take on some projects in addition to your work in QA and perform well, it’s much more likely that they would consider you as a candidate for a full-time BD role. If not, take the skills you’ve learned and apply to BD roles outside of your company.
If you don’t have a BD team in your current company, try to take on some roles outside of your company to help develop the skills you’ll need. Whether it’s volunteering, interning or working for free at a startup or another company that needs the help, take advantage of opportunities outside of your day job. I worked for free at the start of both my first startup internship and job. The company was able to learn more about me and I was able to see if it was a good fit for me.