How does Azure override a MySQL connection string in a php application?
Unless your repository is private you will need to store the wp-config.php file on your Windows Azure instance (or upload it after the fact somehow). Then when the files are pulled into that directory from Github the wp-config.php (assuming no wp-config.php file is in the repo) file will already exist with the connection details.
If you have a private Github repository then it is probably safe to leave the wp-config.php file and values in the repository itself, as only users you give access to the repo can see it.