  In this paper we consider the $q$-deformed totally asymmetric zero range
process ($q$-TAZRP), also known as the $q$-boson (stochastic) particle system,
on the ${\mathbb Z}$ lattice, such that the jump rate of a particle depends on
the site where it is on the lattice. We derive the transition probability for
an $n$ particle process in Bethe ansatz form as a sum of $n!$ $n$-fold contour
integrals. Our result generalizes the transition probability formula by
Korhonen and Lee for $q$-TAZRP with a homogeneous lattice, and our method
follows the same approach as theirs.
