Bianchi type-I Barrow holographic dark energy model in symmetric teleparallel gravity

In this work, we have discussed a spatially homogeneous and anisotropic Bianchi type-I space-time in the presence of Barrow holographic dark energy (infrared cut-off is the Hubble's horizon) proposed by Barrow recently (Physics Letters B 808 (2020): 135643) and matter in the framework of $f(Q)$ gravity where the non-metricity $Q$ is responsible for the gravitational interaction for the specific choice of $f(Q)=\lambda Q^{2}$ (where $\lambda <0$ is a constant). To find the exact solutions to the field equations we consider the deceleration parameter $q$ is a function of the Hubble's parameter $H$ i.e. $q=b-\frac{n}{H}$ (where $b$ and $n$ are constants). We have studied the physical behavior of important cosmological parameters such as the EoS parameter, BHDE and matter density, skewness parameter, squared sound speed, and $\omega _{B}-\omega _{B}^{^{\prime }}$ plane. Also, we constrain the values of the model parameters $b$ and $n$ using $57$ Hubble's parameter measurements.