  In this paper we discuss a relationship between the following two algebras:
(i) the subconstituent algebra $T$ of a distance-regular graph that has
$q$-Racah type; (ii) the $q$-tetrahedron algebra $\boxtimes_q$ which is a
$q$-deformation of the three-point $sl_2$ loop algebra. Assuming that every
irreducible $T$-module is thin, we display an algebra homomorphism from
$\boxtimes_q$ into $T$ and show that $T$ is generated by the image together
with the center $Z(T)$.
