  The $q$-Onsager algebra, denoted $O_q$, is defined by two generators $W_0,
W_1$ and two relations called the $q$-Dolan-Grady relations. Recently,
Terwilliger introduced some elements of $O_q$, said to be alternating. These
elements are denoted $\{{W}_{-k}\}_{k=0}^{\infty},
\{{W}_{k+1}\}_{k=0}^{\infty}, \{{G}_{k+1}\}_{k=0}^{\infty},
\{{\tilde{G}}_{k+1}\}_{k=0}^{\infty}$. The alternating elements of $O_q$ are
defined recursively. By construction, they are polynomials in $W_0$ and $W_1$.
It is currently unknown how to express these polynomials in closed form. In
this paper, we consider an algebra $T_q$, called the quantum torus. We present
a basis for $T_q$ and define an algebra homomorphism $p: O_q \mapsto T_q$. In
our main result, we express the $p$-images of the alternating elements of $O_q$
in the basis for $T_q$. These expressions are in a closed form that we find
attractive.
