So long as Godaddy doesn't block outgoing MySQL connections and your ISP doesn't block incoming MySQL connections, this is technically possible.
You would need to:
- Configure MySQL to listen on your computer's external network interface (i.e. not on a UNIX socket nor on the loopback IP address).
- Ensure that the MySQL service is available on the Internet (which will probably mean configuring your Internet router to perform port forwarding for the MySQL port from the Internet to your development machine)
- Change the connection string in your PHP so it connects to the hostname or IP address of your computer on the Internet (probably your router's external IP address).
You'd probably want to ensure you have a static IP address on the Internet so that you don't need to update the PHP every time your IP changes.
This, however, will be slow (because your MySQL traffic has to go back and forth across the Internet) and risky from a security perspective.
So don't do it. Get a MySQL service from Godaddy.