emacsclient
is an ordinary shell command, so it cannot change the shell's idea of what the current directory is.
You can, however, call emacsclient
like this:
$ cd "$(emacsclient -e '(progn (find-file ....) (print default-directory))')"
i.e., have emacs
print the new directory name when the client returns.
It will not be easy to get right, and the effort is not worth it.
If you are doing more than a simple edit in your emacsclient, you should be using emacs and its shell.