You can create a sub-customer profile in QuickBooks that has a different Ship To address. Let me guide you how to do it.
Go to the Sales menu.
Choose the Customers tab.
Click the New Customer button.
Enter the same customer details except for the Display name as and Ship To address.
Mark the Is sub-customer tick box.
Select the name of the Parent Customer from the drop-down menu.
Make sure to choose the Bill with parent option.
Click the Save button.
Once completed, you can use any of the customer profiles in creating an invoice. It will still be billed to the same customer and billing address. However, it will be delivered to a different Ship-To address.
As of this time, you can only add one Ship To address per customer profile recorded in QuickBooks. I can see how beneficial it is for you and your business to have the ability to add multiple Ship To address. I'd personally share this to our Product Developers for considerations in our future product updates.