  Within the metric formalism of $f(R,T,L_m)$ gravity theories, we investigate
the hydrostatic equilibrium structure of compact stars taking into account both
isotropic and anisotropic pressure. For this purpose, we focus on the
$f(R,T,L_m)= R+ \alpha TL_m$ model, where $\alpha$ is a free parameter. We
derive the modified TOV equations and the relativistic moment of inertia in the
slowly rotating approximation. Using an equation of state (EoS) for
color-superconducting quark stars, we examine the effects of the $\alpha TL_m$
term on the different macroscopic properties of these stars. Our results reveal
that the decrease of the parameter $\alpha$ leads to a noticeable increase in
the maximum-mass values. For negative $\alpha$ with sufficiently small
$\vert\alpha\vert$, we obtain a qualitative behavior similar to the general
relativistic (GR) context, namely, it is possible to obtain a critical stellar
configuration such that the mass reaches its maximum. However, for sufficiently
large values of $\vert\alpha\vert$ keeping negative $\alpha$, the critical
point cannot be found on the mass-radius diagram. We also find that the
inclusion of anisotropic pressure can provide masses and radii quite consistent
with the current observational measurements, which opens an outstanding window
onto the physics of anisotropic quark stars. By comparing the $I-C$ relations
of isotropic quark stars in modified gravity, we show that such a correlation
remains almost unchanged as the parameter $\alpha$ varies from GR counterpart.
On the other hand, given a fixed $\alpha$, the $I-C$ relation is insensitive to
variations of the anisotropy parameter $\beta$ to $\mathcal{O}(4\%)$.
