Essentially there is not much difference between a QA and a Scrum master other than roles and specific duties.
Despite sharing many qualities, keep in mind the Scrum master is a direct link for the analysis team to communicate and receive instruction with the product originator, a role requiring a firm grasp of communication and organization geared towards team efficiency.
A good QA may already have experience micromanaging within their group and shift to overhead supervision of the staff, but even the best need to learn to adapt the instructions passed from higher-ups to be understood by all levels below them.
It may benefit a QA to take courses in leadership skills and start to familiarize themselves with the tools for group intercommunication, as well as presenter skills for future broadcasting to team members the project's goals and general direction. Having these kinds of soft skills really define a role like a scrum master and separates it from a general analysis position with extra authority.
So yes, a QA can become a scrum master, but only if they accept the role of liaison to the main hiring power and the responsibilities that come with it alongside being a top member of the team.