It seems that MySQL support is included in PHP, and, from the Go Daddy website apparently even their cheapest plan includes database access (on their Windows and Linux plans).
So, to setup phpbb what you first have to do is create the database, according to the Database Management guide from
https://www.godaddy.com/gdshop/suppo...se=%2B&ci=1187 how you do this with Go Daddy is:
"1. Log in to your account manager.
2. From Hosting & Email list, select Web Hosting & Databases.
3. On the Manage Hosting Accounts page, click the Databases link next to the hosting account for which you want to manage the databases.
4. You may modify an existing database or create a new database from this page."
So, some how from there you should be able to create a MySQL database and username, and set the username to be able to access and modify the database. Hopefully that page will also have all of the information you're supposed to enter into the form to setup phpbb correctly (like the mysql server host name).