This paper provides extensions of the \(q\)-Hahn boson process and the \(q\)-Hahn TASEP (totally asymmetric simple exclusion process) and exhibits some features of their mutual relation. A Bethe ansatz is used to compute the expectations of a family of observables of the \(q\)-Hahn TASEP, and a Fredholm determinant formula is proven for the distribution of the location of any given particle.