  We investigate two classes of non-minimally coupled curvature-matter models
in the FLRW universe with a perfect fluid and analyze their cosmological
implications using Supernova Ia, Observed Hubble Data, and Baryon Acoustic
Oscillation measurements. Non-minimal coupling is introduced via an additional
term $\int d^4x \sqrt{-g} \mathcal{G}({\cal L}_{m}) f_2(R)$ in the
Einstein-Hilbert action. To obtain observational constraints, we use an
exponential-type fluid-pressure profile $p = p_0e^{ak}$ characterized by the
dimensionless parameter $k$ and parameterize $f_2(R)$ as $R^n$ with another
dimensionless parameter $n$. Two additional parameters, $\alpha$ and $\beta$ in
the functional form of $\mathcal{G}({\cal L}_{m})$ determine the coupling
strength. We identify significant regions in the $(n, k)$-parameter space for
fixed coupling strength values where non-minimally coupled models align with
observed late-time cosmic evolution. Additionally, we explore and discuss
features of energy transfer between the curvature and matter sectors using
observational data.
