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.
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