  Let $d$ and $m$ be two distinct squarefree integers and $\mathcal{O}_K$ the
ring of integers of the quadratic field $K=\mathbb{Q}(\sqrt{d})$. Denote by $
H_K(\alpha, m)$ a quaternion algebra over $K$, where $\alpha\in \mathcal{O}_K$.
In this paper we give necessary and sufficient conditions for $ H_K(\alpha, m)$
to split over $K$ for some values of $\alpha$, and we obtain a complete
characterization of division quaternion algebras $ H_K(\alpha, m)$ over $K$
whenever $\alpha$ and $m$ are two distinct positive prime integers. Examples
are given involving prime Fibonacci numbers.
