The career path most definitely exists if you make it for yourself. I know of a few testers who have gone on to purely security. Functional approaches and security approaches can often overlap each other, and many of those who I speak with in the security field are grateful for their previous test experience.
After a time however, I've found that often functional tests and security tests tend to overlap and blend in with each other in agile-ish environments. About the only time that I actually think of "security" tests is when I've been asked to do penetration tests or run vulnerability scans. Otherwise, it's just testing.
Short answer - absolutely, they can be closely related.