  We show that, for all positive integers $n_1, \ldots, n_m$, $n_{m+1}=n_1$,
and any non-negative integers $j$ and $r$ with $j\leqslant m$, the expression
$$ \frac{1}{[n_1]}{n_1+n_{m}\brack n_1}^{-1}
\sum_{k=1}^{n_1}[2k][k]^{2r}q^{jk^2-(r+1)k}\prod_{i=1}^{m} {n_i+n_{i+1}\brack
n_i+k} $$ is a Laurent polynomial in $q$ with integer cofficients, where
$[n]=1+q+\cdots+q^{n-1}$ and ${n\brack k}=\prod_{i=1}^k(1-q^{n-i+1})/(1-q^i)$.
This gives a $q$-analogue of a divisibility result on the Catalan triangle
obtained by the first author and Zeng, and also confirms a conjecture of the
first author and Zeng. We further propose several related conjectures.
