On quaternion algebras that split over quadratic number fields

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.