In Settings/Notes/Accounts/Add Account I can add a Gmail account (eg email@example.com), but a simple Google account (eg firstname.lastname@example.org) I can not.
No error, or success, message is displayed during and after the process of entering username and password. At the end of the process the account is not added to the list of Notes accounts.
But I was able to add the same email (eg email@example.com) in Notes app under the Mac OS X (version 10.13.6).
Only in iOS (version 10.3.3) I could not do this.