  An accelerated expansion phase is being experienced by the universe due to
the presence of an unknown energy component known as dark energy (DE). To find
out the cosmic evolution scientists ever tried to modify Einstein's
gravitational theory and its unexplored parts. We also look forward to address
the same problem with a different approach based on interaction between matter
and geometry. For this purpose we consider $f(R,T,Q)$ modified theory (where
$R$ is the Ricci Scalar, $T$ is the trace of energy-momentum tensor (EMT)
$T_{uv}$ and $Q=R_{uv}T^{uv}$ is interaction of EMT $T_{\mu\nu}$ and Ricci
Tensor $R_{uv}$). We formulate modified field equations in the background of
Friedmann-Lema$\hat{i}$tre-Robertson-Walker (FLRW) model which is defined as
$ds^2=dt^2-a(t)^2(dx^2+dy^2+dz^2 )$, where $a(t)$ represents the scale factor.
In this formalism energy density is found using covariant divergence of
modified field equations. $\rho$ involves a contribution from non-minimal
matter geometry coupling which helps to study different cosmic eras based on
equation of state (EOS). Furthermore, we apply the energy bounds to constrain
the model parameters establishing a pathway to discuss the cosmic evolution for
best suitable parameters in accordance with recent observations.
