If you want to publish your software using GPL then you have to provide the source code no matter is it free or you charge money for it. In GPL "Free" doesn't mean cost free but freedom to change the code etc.
In your case if you don't wish to give away your source code, then you you should use LGPL license. Qt is available under LGPL license too. The only restriction it puts is that you cannot statically link with Qt libs. You ought to use dlls as you are using. And if you only use Qt dlls then you can even charge money for your application.
http://en.wikipedia.org/wiki/GNU_Lesser_General_Public_License#Differences_from_the_GPL
You can use LGPL library in your commercial, closed source, non-free/free application provided you link to the LGPL library dynamically (using .dlls, .so etc.).