Hi there,
Custom DNS records - like the ones used to generate Google Apps DNS records - can only be used when your name servers are pointing to WordPress.com.
If you've set your domain to use GoDaddy's nameservers, they indeed should be able to allow you to add MX records - or do this on your behalf - in order to get your Google Apps working.
If GoDaddy is still unable to help you make the changes you require, you might want to ask them what name servers you can use so that they will be able to assist you.
Please let me know if this continues to be an issue and we'll continue to try to help sort it out.