Unipotent orbits of elements in a quaternion ring of odd order

Let $n\in \text{NN}$ and let $q=p^r$ be an odd prime power. Let $R$ be a finite commutative local principal ring of cardinality $q^n$ with $R\mathrm{/}J(R)\simeq GF(q)$. We study the conjugation action of the group of all unipotent elements in the quaternion ring $H(R)$ on $H(R)$ and we classify the resulting unipotent similarity classes, using a reduction to the ring of $2$-by-$2$ matrices over $R$.