You could use ImplicitRegion and RegionIntersection
regA = ImplicitRegion[z == x + y + 2 && Between[x, {-10, 10}] && Between[y, {-10, 10}], {x, y, z}];
regB = ImplicitRegion[z == x^2 + 4, {x, y, z}];
Show[
Region[Style[regA, Orange]],
Region[Style[RegionIntersection[regA, regB], Black, Thick]],
Boxed -> True, Axes -> True, BoxRatios -> 1
]
