<p>In this paper, we study the Hyers–Ulam stability of quaternion difference equations with bilateral constant coefficients. We begin by deriving the general solutions of first-order equations using induction, followed by an analysis of their Hyers–Ulam stability through the complex representation of quaternions. The Hyers–Ulam constant on a finite interval is then explicitly derived by establishing the relationship between the solutions of inhomogeneous and homogeneous equations. These results are then extended to high-order quaternion difference equations, and illustrative examples are provided to verify the theoretical analysis.</p>