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 Select ST_Intersection(atable_a.geomcolumn, btable_b.geom)column As intersect_ab
FROM a INNERfrom JOINtable_a b ON ST_Intersection(a,b)
join table_b WHERE
on ST_Overlaps(atable_a.geom,MUKEY b= table_b.geomMUKEY)
;
(...or something very similar...)
More info here