In this paper, a non-Gaussian Ornstein–Uhlenbeck process driven by a Hermite–Ornstein–Uhlenbeck process is introduced, which belongs to the qth Wiener chaos. A systematic procedure to identify the drift parameter θ and the Hurst parameter H is given based on the study of the limit behavior of its quadratic variations. Estimators for these two parameters and their asymptotic properties are studied.