  We investigate the nonequilibrium dynamics of the $S=1$ quantum spin chain
subjected to a time-dependent external drive, where the driving frequency is
adiabatically decreased as a function of time (``Floquet adiabatic
evolution''). We show that when driving the rhombic anisotropy term (known as
the ``two-axis countertwisting'' in the context of squeezed spin states) of a
N\'eel antiferromagnet, we are able to induce an overall enhancement in the
quadrupole polarization, while at the same time suppressing the staggered
magnetization order. The system evolves into a new state with a net quadrupole
moment and antiferroquadrupolar correlations. This state remains stable at long
times once the driving frequency is kept constant. On the other hand, we find
that we cannot achieve a quadrupole polarization for the symmetry-protected
Haldane phase, which remains robust against such driving.
