My research interests are two mathematical objects; the first one is a combinatorial object known as a Q-polynomial distance-regular graph (DRG) and the second is an algebraic object known as the double affine Hecke algebra (DAHA). Q-polynomial DRGs can be thought of as a discrete analogue of rank one symmetric space. The Q-polynomial property appears in many other areas of mathematics, such as Lie theory, quantum algebras, and orthogonal polynomials. The DAHA was introduced by Cherednik at the beginning of 1990. Since then, the theory of DAHA has been receiving great attention and connected to many other areas, such as integrable systems, algebraic geometry, quantum algebras, and orthogonal polynomials. I discovered a relationship between Q-polynomial DRGs and the DAHA of rank one. This connection provides a new approach to the study of the theory of distance-regular graphs and the DAHAs. Using this relationship, I explore connections between Q-polynomial DRGs and the non-symmetric (or Laurent) case of the terminating branch of the Askey scheme of orthogonal polynomials. I also have ongoing research into the generalized Terwilliger algebra and its connections to orthogonal polynomials, mathematical physics, quantum algebra, and Lie theory.