  We derive a working model for the Tolman-Oppenheimer-Volkoff equation for
quark star systems within the modified $f(T, \mathcal{T})$-gravity class of
models. We consider $f(T, \mathcal{T})$-gravity for a static spherically
symmetric space-time. In this instance the metric is built from a more
fundamental tetrad vierbein from which the metric tensor can be derived. We
impose a linear $f(T)$ parameter parameter, namely taking $f=\alpha T(r) +
\beta \mathcal{T}(r) + \varphi$ and investigate the behavior of a linear
energy-momentum tensor trace, $\mathcal{T}$. We also outline the restrictions
which modified $f(T, \mathcal{T})$-gravity imposes upon the coupling
parameters. Finally we incorporate the MIT bag model in order to derive the
mass-radius and mass-central density relations of the quark star within $f(T,
\mathcal{T})$-gravity.
