I am still logged in using a browser X, can CSRF attack work when I use another (default) browser Y on the same machine?
Assume the web server has CSRF vulnerability. No hidden token checking will be done on the server.
No. CSRF works by using a session already in place, and sessions can only exist within the same browser. Or put another way, all current/major browsers do not share sessions.
EDIT: As noted in the comment, it is theoretically possible using flash cookies. IMO, flash cookies are evil to begin with though.
Assuming that the existing CSRF vulnerability is based on the presence of a login cookie, then the simple answer is no (in the general case).
However, there are other CSRF flaws which are no based on cookies, which may be relevant to other browsers.