How can we know the polygon contains many pixels?

Study some math. Qt doesn't help you calculate that, but a fair bit of "Geometry":http://en.wikipedia.org/wiki/Geometry will.

Does "this":http://en.wikipedia.org/wiki/Polygon_triangulation help? Anyway, this is not a Qt related question.

The fact that Qt can fill your polygon, does not mean that it knows how many pixels are red. Filling doesn't work that way.
You could of course paint your polygon into a QImage, and can the image line by line to count the number of colored pixels. Not very efficient, but perhaps enough for your goals?