  We give a description of the Halln\"as--Ruijsenaars eigenfunctions of the
2-particle hyperbolic Ruijsenaars system as matrix coefficients for the order 4
element $S\in SL(2,\mathbb{Z})$ acting on the Hilbert space of $GL(2)$ quantum
Teichm\"uller theory on the punctured torus. The $GL(2)$ Macdonald polynomials
are then obtained as special values of the analytic continuation of these
matrix coefficients. The main tool used in the proof is the cluster structure
on the moduli space of framed $GL(2)$-local systems on the punctured torus, and
an $SL(2,\mathbb{Z})$-equivariant embedding of the $GL(2)$ spherical DAHA into
the quantized coordinate ring of the corresponding cluster Poisson variety.
