I need to compute some statistics of the pixels of a given raster that are coincident to the polygons of a vector file. Is there a simple and effective way to do this in Quantum GIS or similar?
You can use the GRASS plugin within QGIS that has the module v.rast.stats
There was also a mention of the zonal statistics in qgis-trunk last month. I'm not sure if this was added in QGIS 1.7.
I don't know why you are doing that but the number of pixels will depend of your raster resolution and I'm not sure the method proposed by manning will convert automatically the pixels into area (but you should check!).
Vectorizing your raster file and calculating the area overlaps will give you a result in the defined units (meters, miles, or whatever).
if you want to proceed that way, then GRASS will help you. See r.to.vect function.