Let Q_8 be the quaternion group of order 8 and {chi } its faithful irreducible character. Then {chi } can be realized over certain imaginary quadratic number fields K={mathbb Q}bigl (sqrt{-N}bigr ) but not over their rings of integers (Feit, Serre); here N is a positive square-free integer. We show that this happens precisely when {mathbb Q}bigl (sqrt{N}bigr ) but not {mathbb Q}bigl (sqrt{2}, sqrt{N}bigr ) can be embedded into a Q_8-field over the rationals (Galois with group Q_8) and N is not a sum of two integer squares. In particular, we get that {chi } cannot be integrally realized if N is (properly) divisible by some prime qequiv 7,({textrm{mod},}8).