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.
