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The fundamental nature and origin of dark energy are one of the premier mysteries of theoretical physics. In General Relativity Theory, the cosmological constant $Λ$ is the simplest explanation for dark energy. On the other hand, the cosmological constant $Λ$ suffers from a delicate issue so-called fine-tuning problem. This motivates one to modify the spacetime geometry of Einstein's GR. The $f(Q)$ gravity is a recently proposed modified theory of gravity in which the non-metricity scalar $Q$ drives the gravitational interaction. In this article, we consider a linear $f(Q)$ model, specifically $f(Q)=αQ + β$, where $α$ and $β$ are free parameters. Then we estimate the best fit values of model parameters that would be in agreement with the recent observational data sets. We use 57 points of the updated $H(z)$ data sets, 6 points of the BAO data sets, and 1048 points from the Pantheon supernovae samples. We apply the Bayesian analysis and likelihood function along with the Markov Chain Monte Carlo (MCMC) method. Further, we analyse the physical behaviour of cosmological parameters such as density, deceleration, and the EoS parameters corresponding to the constraint values of the model parameters. The evolution of deceleration parameter predicts a transition from decelerated to accelerated phases of the universe. Further, the evolution of equation of state parameter depicts quintessence type behaviour of the dark energy fluid part. We found that our $f(Q)$ cosmological model can effectively describe the late time cosmic acceleration without invoking any dark energy component in the matter part.