We consider the embedding problem for shifts of finite type (SFT) into the Fibonacci-Dyck shift. First, we find the reflection principle does not hold for the path in the Fibonacci-Dyck shift. Then, we obtain the limit of a sequence of topological entropies of a class of irreducible (d − 1)-step SFT of the Fibonacci-Dyck shift, where d(≥ 2) are even integers. This limit provide one of the embedding conditions for the Fibonacci-Dyck shift. Finally, we generalize the result on computing the above limit of a sequence of topological entropies to the Markov-Dyck shift.