Firefox browser wouldn't block gmail access that would be your security software or Total cookie protection is set to high that cookies and tracking from gmail is blocked so access to the account is blocked.

Perhaps.

Do I need to do anything more to save my Mozilla Firefox account? I’d like it to respond when I try to open it on my laptop computer. I have never had this issue before. Thanks.

Ask why would you need Gmail asking you to sign-in to save Mozilla? That's sounds almost like a malware message. Did you check the sender to verify the email address?

Are Mozilla.org and accounts@firefox.com the same? Are they both legitimate? How can I tell?