SetSystemOptions[
  "ReduceOptions" -> "ExhaustiveSearchMaxPoints" -> {10000, 100000}];

Clear[n, M];
n = {a, b, c};
M = {1, 2, 3};
TraditionalForm[Simplify[a  x +  b y +  c z + d == 0]] /.
 Solve[{Abs[n . M + d]/Norm[n] == 2,
   1 <= a <= 10, -5 <= b <= 10, -5 <= c <= 10, a > b > c}, {a, b, c,
   d }, Integers]

SetSystemOptions[
    "ReduceOptions" -> "ExhaustiveSearchMaxPoints" -> {10000, 100000}];

Clear[n, M];
n = {a, b, c};
M = {1, 2, 3};
sol = Solve[{Abs[n . M + d]/Norm[n] == 2,
      1 <= a <= 10, -5 <= b <= 10, -5 <= c <= 10, a > b > c}, {a, b, c,d }, Integers]

TraditionalForm[Simplify[a x + b y + c z + d == 0]] /.
 DeleteDuplicates[sol,
  With[{v1 = Normalize[{a, b, c, d} /. #1],
     v2 = Normalize[{a, b, c, d} /. #2]},
    v1 == v2 || v1 == -v2] &]

r = Reduce[ a x + b y + c z + d == 0 && EuclideanDistance[{1, 2, 3}, {x, y, z}] == 2,
            {a, b, c, d}, Integers]

(a | b | c | d) ∈ Integers && (
    (z == 1 && y == 2 && x == 1 && d == -a - 2 b - c) ||
    (z == 3 && ((y == 0 && x == 1 && d == -a - 3 c) ||
                (y == 2 && ((x == -1 && d == a - 2 b - 3 c) ||
                            (x == 3 && d == -3 a - 2 b - 3 c))) ||
                (y == 4 && x == 1 && d == -a - 4 b - 3 c))) ||
    (z == 5 && y == 2 && x == 1 && d == -a - 2 b - 5 c))

Normal @ Solve[1 <= a <= 5 && -5 <= b <= 5 && -5 <= c <= 5 && a > b > c && r,
               {a, b, c, d}, Integers]

SetSystemOptions[
  "ReduceOptions" -> "ExhaustiveSearchMaxPoints" -> {10000, 100000}];
Clear[n, M];
n = {a, b, c};
M = {1, 2, 3};
TraditionalForm[Simplify[a x + b y + c z + d == 0]] /.
 Solve[{Abs[n.M + d]/Norm[n] == 2,
   1 <= a <= 10, -5 <= b <= 10, -5 <= c <= 10, a > b > c,
   GCD[a, b, c, d] == 1}, {a, b, c, d}, Integers]