Integration with Google Merchant Center is not officially supported by ShipperHQ.
If you are using flat rate or table rate carriers, you can input the same flat rate values into your Google Merchant table. However, Google Merchant Center cannot pull live rates from carriers such as UPS or FedEx. The underlying issue is that Google Merchant Center requires you to enter a fixed rate for shipping while the purpose of a live carrier is to return dynamic rates.
If this service is integral to your store, you can find the average shipping price per order and then add a percentage-based surcharge on your live rate carrier. Google just requires the actual rate to be equal to or higher than the rate entered in the google merchant center. This can be taken further to the extent you need. For example, you could work out the average shipping cost for orders in specific weight ranges, then create a shipping rule to apply surcharges based on weight.