In[360]:= Clear[Q];
Q[j_] := 4 + 3 Floor[1/2 (-2 + j)] + 2 Floor[1/2 (-1 + j)];
j = RandomInteger[{0, 50}];
q = RandomInteger[{0, 50}]; n =.;
eX = 1/Pochhammer[n - j + Q[j] + 1,
     If[Mod[j, 2] == 0, 1,
      2]] (Pochhammer[2 q + 5 j - 3, 3] 1/(q + Q[j]) (1) +
      6 (-1 + 5 j + 2 q)^2 1/(1 + q + Q[j]) (n + -j - q - 0) +
      (60 j + 24 q) 1/(2 + q + Q[j]) (n + -j - q - 1) (n - j - q) +
      8 1/(
       3 + q + Q[j]) (n + -j - q - 2) (n - j - q - 1) (n - j -
         q)) - (If[
     Mod[j, 2] ==
      0, (-4 (-4 + 15 j + 4 q) + 8 Q[j] + (
        6 (1 - 10 j + 25 j^2 - 4 q + 20 j q + 4 q^2))/(
        1 + q + Q[j])) + (-8 + (60 j + 24 q)/(2 + q + Q[j])) (n - j -
         q) +  8/(3 + q + Q[j]) (n - j - q) (n - j - q - 1), -16 + (
      60 j + 24 q)/(2 + q + Q[j]) + 8/(3 + q + Q[j]) (n - j - q)]);
Simplify[eX]

Out[364]= 0