  We consider quantum quenches in the so-called $q$-boson lattice model. We
argue that the Generalized Eigenstate Thermalization Hypothesis holds in this
model, therefore the Generalized Gibbs Ensemble (GGE) gives a valid description
of the stationary states in the long time limit. For a special class of initial
states (which are the pure Fock states in the local basis) we are able to
provide the GGE predictions for the resulting root densities. We also give
predictions for the long-time limit of certain local operators. In the
$q\to\infty$ limit the calculations simplify considerably, the wave functions
are given by Schur polynomials and the overlaps with the initial states can be
written as simple determinants. In two cases we prove rigorously that the GGE
prediction for the root density is correct. Moreover, we calculate the exact
time dependence of a physical observable (the one-site Emptiness Formation
Probability) for the quench starting from the state with exactly one particle
per site. In the long-time limit the GGE prediction is recovered.
