  The $q$-Narayana numbers $N_q(n,k)$ and $q$-Catalan numbers $C_n(q)$ are
respectively defined by $$ N_q(n,k)=\frac{1-q}{1-q^n}{n\brack k}{n\brack
k-1}\quad\text{and}\quad C_n(q)=\frac{1-q}{1-q^{n+1}}{2n\brack n}, $$ where
${n\brack k}=\prod_{i=1}^{k}\frac{1-q^{n-i+1}}{1-q^i}$. We prove that, for any
positive integers $n$ and $r$, there holds \begin{align*}
\sum_{k=-n}^{n}(-1)^{k}q^{jk^2+{k\choose 2}}N_q(2n+1,n+k+1)^r \equiv 0
\pmod{C_n(q)}, \end{align*} where $0\leqslant j\leqslant 2r-1$. We also propose
several related conjectures.
