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We study a relationship between $Q$-polynomial distance-regular graphs andthe double affine Hecke algebra of type $(C^{\vee}_1,C_1)$. Let $\Gamma$ denotea $Q$-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, considerthe corresponding characteristic vector. These characteristic vectors form abasis 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 affinetransformation, (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 dualadjacency matrix of $\Gamma$ with respect to $x$. To obtain our results we usethe theory of Leonard systems.

$Q$-polynomial distance-regular graphs and a double affine Hecke algebra of rank one