Any trick to solve this optimization problem?

You can solve it using the Lagrange multipliers, introducing some slack variables (e1,e2,e3,e4) to handle the inequalities, assuming first to simplify c > 0,(c^2) as follows:

f = c^2 x/y^2 + y/x;
L = f + l1 (x - 1 - e1^2) + l2 (n - x - e2^2) + l3 (y - 1 - e3^2) + l4 (m - y - e4^2)
grad = Grad[L, {x, y, l1, l2, l3, l4, e1, e2, e3, e4}];
sols = Solve[grad == 0, {x, y, l1, l2, l3, l4, e1, e2, e3, e4}];
res = {f, x, y, l1, l2, l3, l4, e1^2, e2^2, e3^2, e4^2} /. sols;
res0 = Union[res];
MatrixForm[res0]

Now in res0 we have the f values at the diverse stationary points as well as the values for e1^2,e2^2,e3^2,e4^2 that should be non negative to be feasible. Here when ek = 0 means that the k constraint is active. We can proceed in the same way in the case of -c^2.

We can further reduce this set to

res1 = {res0[[2]], res0[[5]], res0[[6]], res0[[8]], res0[[9]], res0[[11]], res0[[13]], res0[[14]]};
MatrixForm[res1] 

NOTE

From the results obtained we can observe that the extrema are always at the boundary.