Don't search for a point, search for an area.
Also, you can use each layer of the hierarchy for objects that are larger than their boundaries. In your image that means testing the full-screen level which contains zero objects, move down the hierarchy and test the southeast quadrant which has zero objects, and then because your zone spans two areas you would test against both the southeast and southwest quadrants.
You would also want to test a layer below that, so your green box would search for any children in the main box that contains it because it fits inside each, plus test both northeast and southeast on the grid containing the pink box.
Or in other words:
Quadtree root overlaps, but the collection is empty.
Quadtree [ SE ] overlaps, but the collection is empty.
Quadtree [ SE ] [ SE ] overlaps, and has one item in the collection for testing.
Quadtree [ SE ] [ SE ] [ SE ] overlaps, but node is null so cannot descend
Quadtree [ SE ] [ SE ] [ SW ] overlaps, but node is null so cannot descend
Quadtree [ SE ] [ SE ] [ NE ] overlaps, but node is null so cannot descend
Quadtree [ SE ] [ SE ] [ NW ] overlaps, but node is null so cannot descend
Quadtree [ SE ] [ SW ] overlaps, and has one item in the collection for testing
Quadtree [ SE ] [ SW ] [ NE ] overlaps, but node is null so cannot descend
Quadtree [ SE ] [ SW ] [ SE ] overlaps, but node is null so cannot descend
Again, the goal of a quadtree for spatial operations is to reduce the number of tests. A level might have thousands or tens of thousands of objects, but with careful level design you can reduce any spatial query to a single-digit number of items.
If you have a strict quadtree the pink object should not exist across the boundaries, but should be placed in the layer above. That is, instead of existing in both [SE][SW] and [SE][SE], it should only exist in [SE] since that fully contains it. A loose quadtree allows objects to overlap somewhat, often by 25% (half of a child node). It requires more tests across neighbors, but means fewer graph changes when objects move.