We derive a working model for the Tolman–Oppenheimer–Volkoff equation for quark star systems within the modified $$f(T, \mathcal {T})$$ f ( T , T ) -gravity class of models. We consider $$f(T, \mathcal {T})$$ f ( T , 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, namely taking $$f=\alpha T(r) + \beta \mathcal {T}(r) + \varphi $$ f = α T ( r ) + β T ( r ) + φ and investigate the behaviour of a linear energy-momentum tensor trace, $$\mathcal {T}$$ T . We also outline the restrictions which modified $$f(T, \mathcal {T})$$ f ( T , 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 $$f(T, \mathcal {T})$$ f ( T , T ) -gravity.