Do I need another account to add a password to Google sign-in or link Google to an email account?

No. Sign in to your existing account and open its sign-in settings. If you normally use Google, verify your current email as prompted and set a password. If you already use email and password, confirm that password and choose the Google account to link. Before unlinking Google, make sure you can still sign in with email and password. Do not send passwords or verification codes to Support.