A new method for generating q-Markov Covers for SISO discrete-time systems is proposed. It is based on computing first the impulse-response Gramian from the Markov parameters and covariances, and then solving the Lyapunov equation inversely to get the system matrices in controllability canonical form. The conditions for existence of q-Markov Covers are also derived. The method is illustrated by a numerical example and is shown to be computationally simple.