How can I determine if a certain polygon falls within another separate polygon using QGIS?
Another option is to use the Spatial Query tool from the Vector menu.
Enter your source layer & reference/ intersect layer, and specify "within" as the criterion. If you want to only run the query using selected features of either layer, select these before running the query.
If you wish, you can then use the button shown below to create a new layer from the resulting selection.
The new layer is then added to the table of contents window. You can then right click on the layer name and "save as" if desired.
Add your polygon layers and then go to Vector>Research Tools>Select By Location. Set your parameters accordingly and make sure to check the 'Within' box for the Geometric predicate, and run it.