In this present manuscript, by applying fractional quantum calculus, we study a nonlinear Duffing type equations with three sequential fractional $q$-derivatives. We prove the existence and uniqueness results by using standard fixed point theorems (Banach and Schaefer fixed point theorem). We also discuss the Ulam-Hyers and the Ulam-Hyers-Rassias stabilities of mentioned Duffing problem. Finally, we present an illustrative example and nice application, Duffing-type oscillator equation, for regarding our outcomes.
