  This paper explores the dark energy phenomenon within the context of
$f(R,L_m)$ gravity theory. Two specific non-linear $f (R, L_m)$ models are
considered: $f(R,L_m)=\frac{R}{2}+L_m^\alpha$ and
$f(R,L_m)=\frac{R}{2}+(1+\alpha R)L_m$, where the parameter $\alpha$ is free.
Here, we adopt a parametrization form for the Hubble parameter in terms of
redshift $z$ as $H(z)=H_0 \left[A(1+z)^3+B+\epsilon
\log(1+z)\right]^\frac{1}{2}$, which allows for deviations from the standard
$\Lambda$CDM model at both low and high redshifts. We then incorporate the
Hubble parameter solution into the Friedmann equations for both models. We
employ Bayesian analysis to estimate the constraints on the free parameters
$H_0$, $A$, $B$, and $\epsilon$ using the Hubble measurements and the Pantheon
dataset. Further, we investigate the evolution of key cosmological quantities,
such as the deceleration parameter, energy density, pressure, EoS parameter,
and energy conditions. The evolution of the deceleration parameter reveals a
significant transition from a decelerating phase to an accelerating phase in
the Universe. The EoS parameter exhibits quintessence-like behavior for both
non-linear $f (R, L_m)$ models.
