In [7] the $q$ tetrahedron algebra $\boxtimes_{q}$ was introduced
as a $q$ analogue of the universal enveloping algebra of the
three point loop algebra $\mathit{sl}_{2} \otimes \mathbf{C}[t,t^{-1},(t-1)^{-1}]$.
In this paper the relation between finite dimensional $\boxtimes_{q}$
modules and finite dimensional modules for $U_{q}(L(\mathit{sl}_{2}))$,
a $q$ analogue of the loop algebra $L(\mathit{sl}_{2})$, is studied.
A connection between the $\boxtimes_{q}$ module structure
and $L$-operators for $U_{q}(L(\mathit{sl}_{2}))$ is also discussed.