  In this study, we analyze the dynamics of the interaction between dark matter
and curvature-driven dark energy in viable $f(R)$ gravity models using the
framework of dynamical system analysis. We incorporate this interaction by
introducing a source term in their respective continuity equations, given by $Q
= \frac{\kappa^2 \alpha}{3H}\tilde{\rho}_{\rm m}\rho_{\rm curv}$, and examine
two $f(R)$ gravity models that comply with local gravity constraints and
cosmological viability criteria. Our findings reveal subtle modifications to
fixed points and their stability criteria when compared to the conventional
dynamical analysis of $f(R)$ gravity models without matter-curvature
interactions as proposed and examined in prior literature. We determine the
parameter limits associated with the stability criteria of critical points of
the dynamical system for both the models. Additionally, the introduction of
interaction reveals variations in the dynamical aspects of cosmic evolution,
contingent on the range of values for the relevant model parameters. These
results are consistent with the observed features of cosmic evolution within
specific limits of the $f(R)$ models parameters and the coupling parameter
$\alpha$. We also examine the evolutionary dynamics of the universe in the
interacting scenario through cosmological and cosmographic parameters. Our
analysis demonstrates that the interacting scenario comprehensively accounts
for all the observed phases of the universe's evolution and also results in a
stable late-time cosmic acceleration. Additionally, we've studied how the
coupling parameter affects evolutionary dynamics, particularly its impact on
the matter-to-curvature energy density ratio. Our findings suggest that the
chosen form of interaction can also address the cosmic coincidence problem.
