Summary: The Markov-Dyck shifts arise from finite directed graphs. An expression for the zeta function of a Markov-Dyck shift is given. The derivation of this expression is based on a formula in [G. Keller, J. Comb. Theory, Ser. A 56, No. 1, 75–83 (1991; Zbl 0718.94013)]. For a class of examples that includes the Fibonacci-Dyck shift the zeta functions and topological entropy are determined.