No this is not currently possible. It may be possible for a plugin to do that, maybe a layer that allows users to select their language, stores it in a cookie, then possibly a process module that checks for the cookie and changes which language files are loaded.
But I think this is a rare case. Nearly all Q2A sites would be one language because people would want to ask and answer in one language. There wouldn't be a way for one user to post a question in English and for another to view it in Korean.