How can I contact blizzard support without login to my account

View their contact page here

You can call them at this number according to this site.

United States/Canada   1-800-592-5499

Mobile Authenticator

Like fbueckert said in their comment, if it's a Mobile Authenticator, try closing and re-opening it.

Another possibility is that your Serial has got out of Sync. Which this means you will most definitely have to get in contact with customer support.

Suggestion

Since you can't log into your account, I'd recommend giving them a call. I've had both good and bad experiences while calling Battle.net/Blizzard.

North America

Hours: 7 am - 8 pm Pacific Time, 7 days a week

1-800-592-5499

Here is the full list of contact information