Merge intersecting polygons into one which are part of the same feature

Why don't you:

1. Select the two polygons you want to intesect
2. Enable Editing
3. Edit-> Merge Selected Features
4. Save edits

There's a couple of ways of going about this but you probably want to dissolve the features (Vector->Geoprocessing Tools->Dissolve). With dissolve you don't need to select anything first as it is all done from the attributes. So, let's say you have a field called 'Type' (for example). Then in your example your polygons would all be type 'A' (and you could have some other polygons of type 'B' which you don't want to join with ones of type 'A'). If you select the 'Type' field (or other appropriate attribute in your case) as the attribute to dissolve on then all features of the same type will join together (internal boundaries will be dropped). This may well result in a multipolygon (a single feature which has more than one part, like an archepelago), which might or might not be acceptable. If it is undesirable then simply split the feature up into its constituent polygons by using Vector->Geometry Tools->Multipart to Singlepart.

The alternative is to buffer your polygons by zero, but this probably will join polygons of different types together where you don't want it to happen, so dissolve (optionally follwed by multipart to single part) is safer.