Create a new layer from overlap between two layers?

You can do it with the Vector tools built into the newer versions of QGIS.

I am using 1.8 at the moment, and it has a Vector menu.

Got to: Vector > Geoprocessing Tools > Intersect.

Alternatively, you can probably devise query on the attribute table that will select just the features you want. Once selected you can export that to another layer.

If you use the GRASS GIS toolbox within QGIS, then you can do the job with the v.overlay tool. For a quickstarter, see http://grass.osgeo.org/wiki/QGIS_GRASS_Cookbook