Our analysis focuses on the Generalized Chaplygin Gas (GCG) model within the $f(Q, L_{m})$ gravity framework, assuming $f(Q,L_{m})=\beta Q+\delta L_{m}$ with $L_{m}=-\rho $ . Using the GCG equation of state $p=-\frac{A}{\rho ^{\alpha }}$ , we derive expressions for energy density $\rho (z)$ and the Hubble parameter $H(z)$ . Constraining parameters through MCMC analysis with 31 cosmic chronometers, 15 BAO points, recent DESI DR2 BAO points and 1701 Pantheon+, we find best-fit values $H_{0}=74.026^{+3.332}_{-3.317}$ km/s/Mpc, $A_{s}=0.880^{+0.019}_{-0.020}$ and $\alpha =-0.001^{+0.053}_{-0.052}$ which are consistent with local measurements. The deceleration parameter transitions at $z_{tr} \approx 0.79$ , with present value $q_{0}=-0.61$ , while the equation of state evolves toward $\omega =-1$ with $\omega _{0} \approx -0.79$ . Energy conditions are satisfied except for the SEC, which is violated during acceleration. The model predicts a cosmic age of 13.42 Gyr and shows freezing quintessence behavior in the $\omega -\omega '$ plane, confirming its potential as a viable dark energy candidate.
