If you really mean "forget the password" it probably already did within microseconds of you entering it.
Persistence of authentication through the login session is maintained in Ubuntu-ish systems by `ssh-agent` and `gnome-keyring-daemon`. By their nature of operation (non-invertable hashing) it may be fundamentally impossible to selectively remove one authentication.
As you note, logging out destroys the cached authentication, `ssh_agent -k` would kill the cache without logging out (but other things would fail to authenticate too).
This looks like you can have single-sign-on ease or fine-grained authentication control, pick one.