so that I can query at points (Gx_q, Gy_q, Gz_q) and find their (x,y,z) inputs that give G.
But there may exist no such point or many of them. Suppose for example that Gx=1,Gy=1,Gz=1 everywhere in x,y,z. Then if Gx_q=5, Gx_q=6,Gz_q=7 there will exist no (x,y,z) agreeing with this query point for any reasonable interpolator. If, on the other hand Gx_q=1, Gx_q=1,Gz_q=1 then all (x,y,z) agree with it.
The best I think you can do is solve for one coordinate (x,y,z) at which the interpolation of G is closest to (Gx_q, Gy_q, Gz_q) according to some error metric. That can be approached as an inverse problem as in the following example:
Err = abs(Gx(:)-Gx_q)+abs(Gy(:)-Gy_q)+abs(Gz(:)-Gz_q);
xyzInitial = [x(i), y(i), z(i)];