  We consider a generalization of the quaternion ring $\Big(\frac{a,b}{R}\Big)$
over a commutative unital ring $R$ that includes the case when $a$ and $b$ are
not units of $R$. In this paper, we focus on the case
$R=\mathbb{Z}/n\mathbb{Z}$ for and odd $n$. In particular, for every odd
integer $n$ we compute the number of non-isomorphic generalized quaternion
rings $\Big(\frac{a,b}{\mathbb{Z}/n\mathbb{Z}}\Big)$
