Yes, a GUI designer can become an interaction designer. However, this process can take years. The length of time involved is analogous to a machine operator trying to become an engineer.
As a GUI designer, you're a technician, while a UX designer creates features that help to bring in business to the company. So there are two ways of becoming an Interaction/UX designer: 1.) Apply for highly competitive UX entry-level roles, such as in Bloomberg. OR 2.) Do side projects, start your own self-funded company and build track record that way. Also, join IxDA if you can. You'll get a lot of support.
Hope that helps!