Google+ recently added a new feature that allows users to assign new username and password for their Google plus page. If you assigned, you can sign in directly to your page instead of going through another account that owns or manages the page. And also it allows to use hangout apps, YouTube, Gmail and other third party tools.
- Open your Google+ page click on the Settings
- Then click Set up a Password button in Third party section
- Google asks to verify your account again with your password, then new window will open
- Google plus automatically generated your user name, it will be like firstname.lastname@example.org
- Enter your new password then click Continue button.
That's all you have set a new username and password to your Google+ page. If you want to set your own user name rather than auto generated username, click on the upgrade to Gmail link shows on above image and then change your own username.