Yes, indeed! FindInstance[ x^3 + 109 x^2 + 224 x - y^2 == 0 && x > 0 && y > 0, {x, y}, Integers] gives {x,y}= {4,5} as a solution. I didn't know that RandomSeeding was available for FindInstance. Thank you sharing your solution Mariusz!
One solution:
FindInstance[y^2 == x^3 + 109 x^2 + 224 x && x > 0 && y > 0, {x, y}, Integers, 1, RandomSeeding -> Automatic] (* {{x -> 4, y -> 52}} *)
