The quadratic has a rational root only if sqrt(b*b - 4ac) is rational.
Since (b*b - 4ac) is an integer, its square root can be rational only if (b*b - 4ac) is a perfect square.
Consider mod 8.
Since b is odd, b*b = 1 (mod 8)
Since a and c are odd, 4ac = 4 (mod 8)
Therefore, (b*b - 4ac) = 5 (mod 8)
But all perfect squares = 0 or 1 or 4 (mod 8).
Therefore, (b*b - 4ac) is not a perfect square, so its square root is not rational and the quadratic has no rational roots.