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In this manuscript, we analyze the viscous fluid cosmological model in the framework of recently proposed $f(Q)$ gravity by assuming three different forms of bulk viscosity coefficients, specifically, $(i)ζ=ζ_{0}+ζ_{1}\left( \frac{\dot{a}}{a}\right) +ζ_{2}\left( \frac{\ddot{a}}{\dot{a}}\right) $, $(ii)ζ=ζ_{0}+ζ_{1}\left( \frac{\dot{a}}{a}\right)$, and $(iii)ζ=ζ_{0}$ and a linear $f(Q)$ model, particularly, $f(Q)=αQ$ where $α\neq 0$ is free model parameter. We estimate the bulk viscosity coefficients and the model parameter values using the combined H(z)+Pantheon+BAO data set. We study the asymptotic behavior of our cosmological bulk viscous model by utilizing the phase space method. We find that corresponding to all three cases, our model depicts the evolution of the universe from matter dominated decelerated epoch (a past attractor) to a stable de-sitter accelerated epoch (a future attractor). Further, we study the physical behavior of effective pressure, effective equation of state (EoS), and the statefinder parameters. We find that the pressure component in the presence of bulk viscosity shows negative behavior and the effective EoS parameter predicts the accelerated expansion phase of the universe for all three cases. Moreover, we obtain that the trajectories of our model lie in the quintessence region and it converges to the $Λ$CDM fixed point in the far future. We find that the accelerated deSitter like phase comes purely from the $\barζ_0$ case without any geometrical modification to GR. Moreover, we find that the late-time behavior of all three cases of viscosity coefficients are identical.