Observationally Constrained Cosmological model in $f(Q,\mathcal{L}_{m})$ Gravity with $H(z)$ parameterization

In the present work, we explore an observationally constrained cosmological model in the framework of $f(Q,\mathcal{L}_{m})$ gravity, where $Q$ denotes the non-metricity scalar and $\mathcal{L}_{m}$ represents the matter Lagrangian density. To derive the modified Friedmann field equations, we consider a flat FLRW space-time. We have considered a specific parameterization of the Hubble parameter $H(z)$ to explore the cosmic evolution, which successfully describes the shift of the cosmos from its initial decelerated expansion period to the current accelerated scenario. The free model parameters are constrained using recent observational datasets including Cosmic Chronometers (CC), Pantheon+SH0ES, Union 3.0, DESI-BAO, and CMB distance priors using MCMC approach through the $\chi^2$-minimization process. The derived results indicate that the present model remains consistent with recent cosmological observations. We note that the deceleration parameter exhibits a signature flipping behavior at transition redshift $z_t \approx 0.643$, confirming the transition from matter-dominated deceleration to dark-energy-driven acceleration. The equation of state (EOS) parameter remains in the quintessence region and exhibits an asymptotical approach to the $\Lambda$CDM limit at late times. Moreover, the estimated cosmic age can be found as $13.724^{+0.087}_{-0.048}$ Gyr, which agrees well with recent observational estimations. The statefinder and Om diagnostics support the quintessence nature of the model. At the same time, the examination of energy conditions reveals that two specific energy conditions, viz. Null Energy Condition (NEC) and Dominant Energy Condition (DEC) are fulfilled, while the Strong Energy Condition (SEC) is violated, validating the accelerated expansion of the universe.