  We study a relationship between $Q$-polynomial distance-regular graphs and
the double affine Hecke algebra of type $(C^{\vee}_1,C_1)$. Let $\Gamma$ denote
a $Q$-polynomial distance-regular graph with vertex set $X$. We assume that
$\Gamma$ has $q$-Racah type and contains a Delsarte clique $C$. Fix a vertex $x
\in C$. We partition $X$ according to the path-length distance to both $x$ and
$C$. This is an equitable partition. For each cell in this partition, consider
the corresponding characteristic vector. These characteristic vectors form a
basis for a $\mathbb{C}$-vector space ${\bf W}$.
  The universal double affine Hecke algebra of type $(C^{\vee}_1,C_1)$ is the
$\mathbb{C}$-algebra $\hat{H}_q$ defined by generators $\{t^{\pm1}_n\}^3_{n=0}$
and relations (i) $t_nt_n^{-1}=t_n^{-1}t_n=1$; (ii) $t_n+t_n^{-1}$ is central;
(iii) $t_0t_1t_2t_3 = q^{-1/2}$. In this paper, we display an
$\hat{H}_q$-module structure for ${\bf W}$. For this module and up to affine
transformation, (i) $t_0t_1+(t_0t_1)^{-1}$ acts as the adjacency matrix of
$\Gamma$; (ii) $t_3t_0+(t_3t_0)^{-1}$ acts as the dual adjacency matrix of
$\Gamma$ with respect to $C$; (iii) $t_1t_2+(t_1t_2)^{-1}$ acts as the dual
adjacency matrix of $\Gamma$ with respect to $x$. To obtain our results we use
the theory of Leonard systems.
