In[1]:=Reduce[Exists[{s, t}, x == s^2 - t^2 && y == s + t && z == s^2 + 3 t], {z}]

Out[1] = (y == 0 && x == 0) || (y != 0 && z == (x^2 - 6 x y + 2 x y^2 + 6 y^3 + y^4)/(4 y^2))