We consider a Markov process on non-negative integer arrays of a certain shape, this shape being determined by general parameters in the model which correspond to drifts. In the case where these drifts are trivial, the arrays are reverse plane partitions. This model is closely related to the Whittaker functions of the quantum group $U_q(\mathfrak{sl}_{r+1})$ and the Toda lattice. Moreover, the Markov process we introduce can be viewed as a one-parameter deformation of the discrete Whittaker processes introduced by Neil O'Connell in [11]. We show that the $q$-deformed Markov process has non-trivial Markov projections to skew shapes.