When you create an account you either have to have a password for your pokemon trainer account, or log in through your google account, which would mean your google username and password. This means you can’t start without having a password.
If these options aren’t enough, you can probably find a relevant topic or email address to contact Pokemon Go support on their website.
*on a side note. As long as you can provide Niantic with the info registered to the account they should be able to recover it (i.e. email adress, original user name, etc). But yeah, if you are registered with Google it shouldn’t be having problems.