Let $𝕜$ be an algebraically closed field of characteristic $p\ne 2$, and let $Q_8$ be the quaternion group. We describe the structures of all simple modules over the quantum double $D\left(𝕜Q_8\right)$ of group algebra $𝕜Q_8$. Moreover, we investigate the tensor product decomposition rules of all simple $D\left(𝕜Q_8\right)$-modules. Finally, we describe the Grothendieck ring $G_0\left(D\left(𝕜Q_8\right)\right)$ by generators with relations.