Wolfram|Alpha does not say "this is the only possible integer solution", and you did not ask for all solutions. Mathematica can find more:
In[121]:= FindInstance[
a + b == 2 (c + d) && 2 a b == c d && a == 1 && b > 0, {a, b, c,
d}, Integers, 4]
Out[121]= {{a -> 1, b -> 33, c -> 6, d -> 11}, {a -> 1, b -> 33,
c -> 11, d -> 6}, {a -> 1, b -> 45, c -> 5, d -> 18}, {a -> 1,
b -> 45, c -> 18, d -> 5}}
