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The paper presents Barrow holographic dark energy (infrared cut-off is the Hubble horizon) suggested by Barrow recently (Physics Letters B 808 (2020): 135643) in an anisotropic Bianchi type-I Universe within the framework of $% f\left( Q\right) $ symmetric teleparallel gravity, where the non-metricity scalar $Q$ is responsible for the gravitational interaction. We consider two cases: Interacting and non-interacting models of pressureless dark matter and Barrow holographic dark energy by solving $f\left( Q\right) $ symmetric teleparallel field equations. To find the exact solutions of the field equations, we assume that the time-redshift relation follows a Lambert function distribution as $t\left( z\right) =\frac{mt_{0}}{l}g\left( z\right) $, where $g\left( z\right) =LambertW\left[ \frac{l}{m}e^{\frac{l-\ln \left( 1+z\right) }{m}}\right] $, $m$ and $l$ are non-negative constants and $t_{0}$ represents the age of the Universe. Moreover, we discuss several cosmological parameters such as energy density, equation of state (EoS) and skewness parameters, squared sound speed, and $(ω_{B}-ω_{B}^{^{\prime }})$ plane. Finally, we found the values of the deceleration parameter (DP) for the Lambert function distribution as $q_{(z=0)}=-0.45$ and $q_{(z=-1)}=-1$ which are consistent with recent observational data, i.e. DP evolves with cosmic time from initial deceleration to late-time acceleration.