  In the framework of a 3-3-1 model with a minimal scalar sector we make a
detailed study concerning the implementation of the PQ symmetry in order to
solve the strong CP problem. For the original version of the model, with only
two scalar triplets, we show that the entire Lagrangian is invariant under a
PQ-like symmetry but no axion is produced since an U(1) subgroup remains
unbroken. Although in this case the strong CP problem can still be solved, the
solution is largely disfavored since three quark states are left massless to
all orders in perturbation theory. The addition of a third scalar triplet
removes the massless quark states but the resulting axion is visible. In order
to become realistic the model must be extended to account for massive quarks
and invisible axion. We show that the addition of a scalar singlet together
with a Z_N discrete gauge symmetry can successfully accomplish these tasks and
protect the axion field against quantum gravitational effects. To make sure
that the protecting discrete gauge symmetry is anomaly free we use a discrete
version of the Green-Schwarz mechanism.
