It would depend on where your MySQL instance is provisioned.
If you’re using the MySQL Render example, it would be available on the hostname shown on the Private Service dashboard (under the “Connect” button, top right). Another consideration for a Private Service is that it would need to be in the same account and region as the calling service.
So that we can take a look at your account/services feel free to raise a ticket from the “Help” or “Contact Support” links while logged into the Dashboard.