The account is logged in on Safari, but I don't know the password.
Is there any possible way to save the password and get Safari to autofill the password when I log out and fill the username?
No, there is no possibility to retrieve the password at this stage. The session state (i.e. the fact that "the account is logged") is stored by the browser in a cookie. The password used to login is not stored anywhere (unless the user explicitly permitted the password to be saved, which is not the case you asked about).
This is a basic security feature protecting anyone from having their password compromised if they left the device unattended for a moment. For the same reason you cannot change the password using Facebook account settings without providing the old one.