Here is a way:
f1[x_, y_] := x + y;
f2[x_, y_] := x^2 + y^2;
pl = Show[
Plot3D[f1[x, y], {x, -1, 1}, {y, -1, 1},
MeshFunctions -> {f2[#1, #2] - f1[#1, #2] &}, Mesh -> {{0}},
MeshStyle -> Thick, BoundaryStyle -> None],
Plot3D[x^2 + y^2, {x, -1, 1}, {y, -1, 1}, Mesh -> None,
BoundaryStyle -> None]]
grCmplx = Normal@Cases[pl, _GraphicsComplex, All];
Cases[grCmplx, _Line, All]
But you may rather try Solve
.