  We quantize a homogeneous and isotropic universe for two models of modified
teleparallel gravity, wherein an arbitrary function of the boundary term,
namely $B$, is present in the action and in the other model a scalar field that
is non-minimally coupled to both the torsion and boundary term. In this regard,
we study exact solutions of both the classical and quantum frameworks by
utilizing the corresponding Wheeler-DeWitt (WDW) equations of the models. To
correspond to the comprehensive classical and quantum levels, in the second
model, we propose an appropriate initial condition for the wave packets and
observe that they closely adhere to the classical trajectories and reach their
peak. We quantify this correspondence using the de-Broglie Bohm interpretation
of quantum mechanics. According to this proposal, the classical and Bohmian
trajectories coincide when the quantum potential vanishes along the Bohmian
paths. Furthermore, we apply the de-parameterization technique to our model in
the realm of the problem of time in quantum cosmological models based on the
WDW equation, utilizing the global internal time denoted as $\chi$, which
represents a scalar field.
