Let $p$ be a prime number, and $F$ be the quadratic real field $\mathbb{Q}{\sqrt{p}}$. Up to isomorphism, there is a unique quaternion $F$-algebra $H:=H_{\infty_1, \infty_2}$ that is ramified at the two finite places of $F$ and splits at all finite places. We compute the class number $h^1(H, \mathcal{O})$ of the normal one group of $H$ with respect to the open compact subgroup $\widehat\mathcal{O}^1$ for a maximal order $\mathcal{O} \subset H$. For certain type of maximal orders, this class number is equal to the number of isomorphism classes of principally polarized abelian surfaces over $\mathbb{F}_p$ in the isogeny class corresponding to the Weil $p$-numbers $\pm \sqrt{p}$. This is a joint work with Chia-Fu Yu.