Only, you'd have to gpl your code
This is not true. You can GPL your code if you want but it is not a requirement. Your code automatically comes under the GPL only when it modifies a program that is already GPL'd. You would have to provide a link to the GPL part of the code, i.e. the QT website (generally in the comments at the beginning of the program), but your code is your code. One of the mobile phone manufactures, I think it is Nokia but I'm not sure, uses QT but has had a closed operating system for years. If it is Nokia, they have recently open-sourced, but it was closed for years even though the QT tool kit was used.