The UPS module doesn't use CURL, but I bet GoDaddy's proxy server is the problem. I don't have any idea what I can do about that though.
...
Oho! There's a comment in drupal_http_request() under the section about https:
<?php
// Note: Only works for PHP 4.3 compiled with OpenSSL.
?>On your GoDaddy site, go to admin/logs/status/php and search for '--with-openssl' on the page. I'll be surprised if it's there.
Joined: 08/07/2007