There's no need for any global objects all the more if you only have one active connection for your application but it's also valid if you are managing multiple connections. The QSqlDatabase class already provides everything you need.
Like written in the documentation, if you are only using one connection, don't give it a name, it will become the default connection and all the SQL related classes will use it by default.
If you have several connections, it's easy to retrieve the one you are interested in by using QSqlDatabase::database.
On a side note, you should properly test the outcome of your open call. Right now it could be failing and you don't care about that possibility.