You don't need Visual Studio, you only need it if you want to use nmake. GCC is perfectly capable on its own if you don't want to use Visual Studio.
qmake builds makefiles, it doesn't build anything else. You need to use your favourite make application to actually do the building. Eg. mingw32-make, or nmake.
Alternatively, you can just use the QtCreator IDE to do all this for you, and then copy the generated exe and appropriate dlls to your distribution directory, then pack up that dir and distribute.
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