<p>Barrow holographic dark energy (BHDE) has recently emerged as a prominent alternative to the standard holographic dark energy paradigm, attracting significant interest in cosmological research. In this work, we investigate the evolution of a universe filled with a mixture of BHDE and pressureless cold dark matter in an anisotropic background within the framework of the Brans-Dicke theory of gravity with a time dependent cosmological constant. The Hubble horizon is chosen as the IR cut-off. The solution of the field equations are obtained by considering a hybrid expansion law for the average scale factor. The dynamical behavior of several cosmological parameters such as the Hubble parameter, deceleration parameter, equation of state parameter etc. are analyzed in detail by plotting their evolution graphs for redshift <i>z</i>. The evolution of some physical parameters and cosmographic parameters are studied. We also examine the energy conditions. We find that the model shows quintessence like or phantom like behavior depending on the Barrow exponent choice. The model also exhibits transition from decelerating phase of cosmic expansion into an accelerating one.</p>