Here is a tutorial for doing the join in QGIS.
To do this in PostGIS, you would create a query or view that does the join between the two tables based on the ST_Intersects function. There are examples of that here, one of which would read:
SELECT ST_Intersection(a.geom, b.geom) As intersect_ab
FROM a INNER JOIN b ON ST_Intersection(a,b)
WHERE ST_Overlaps(a.geom, b.geom)
;