In this paper, we present a quantum property testing algorithm for recognizing a context-free language that is a concatenation of two palindromes $L_{REV}$. The query complexity of our algorithm is $O(\frac{1}{\varepsilon}n^{1/3}\log n)$, where $n$ is the length of an input. It is better than the classical complexity that is $Θ^*(\sqrt{n})$. At the same time, in the general setting, the picture is different a little. Classical query complexity is $Θ(n)$, and quantum query complexity is $Θ^*(\sqrt{n})$. So, we obtain polynomial speed-up for both cases (general and property testing).