Level
may provide a means of getting started. Level breaks down your polynomials into their constituent parts, with various "levels" of complexity. As @Rex Kerr noted in a comment, Level 1 happens to be interesting in the present example.
a=(G u^2 (6 p (2 h+p)-8 (h+p) u+3 u^2))/(12 h^2);
b=(G (3 h+3 p-2 u) u^2)/(3 h^2);
Intersection[Level[a,1],Level[b,1]]