I do a lot of cross-platform development (Mac, Windows, Linux), and for some projects use Qt. It is a fine framework, and provides a rich class library. If you need to deploy on multiple platforms, cannot afford to spend the time/effort on platform-specific front-ends, or the "generic" support for each platform is sufficiently good, then use Qt.
However, Qt inevitably suffers in some ways from the lowest common denominator syndrome, and sometimes does not feel quite native enough. There are also certain features that are either difficult to support, or are simply not provided in the Qt libraries. So if you can afford the time and effort, or your app really demands the attention to detail and fit & finish, then developing separate front-ends may be worth it.
In either case, you ought to be writing your back-end (aka domain) code in a platform-neutral and front-end neutral manner. This way, the front-end is easily replaced, or modified between platforms.
You could always start with a Qt front-end and go for a quick time to market, then develop a native front-end down the line.
In practice, I've noticed that a Qt app on Windows looks most "native", while on Mac there are certain subtle telltale signs that make it look/feel not quite right. And Mac users tend to have much higher expectations when it comes to UI/UX!