Q = Table[Table[, S], S];
For[i = 1, i <= S, i++,
  For[j = 1, j <= S, j++,
    Q[[i, j]] =  (* Numerical calculation that depends on both `i` and `j`*);
    ] (* End For *);
  ] (* End For *) ;

Q = Table[Table[, S], S]; (* Probably this is no longer needed *)
Q = ParallelTable[
(* Numerical calculation that depends on both `i` and `j`*)
,
   {i, 1, S}, {j, 1, S}];