  We propose novel triple-${\bm q}$ multipole orders as possible candidates for
the two distinct low-temperature symmetry broken phases in quadrupolar system
PrV$_2$Al$_{20}$. An analysis of the experiment under [111] magnetic fields
indicates that the {\it ferro} octupole moments in the lower temperature phase
arise from the {\it antiferro} octupole interactions. We demonstrate that the
triple-${\bm q}$ multipole orders can solve this seemingly inconsistent issue.
Anisotropies of quadrupole moments stabilize a triple-${\bm q}$ order, which
further leads to the second transition to a coexisting phase with triple-${\bm
q}$ octupole moments. The cubic invariant of quadrupole moments formed by the
triple-${\bm q}$ components and the characteristic couplings with the octupole
moments in their free energy play important roles. We analyze a multipolar
exchange model by mean-field approximation and discuss the temperature and
magnetic field phase diagrams. Many of the microscopic results, such as the
number of phases and the magnitudes of critical fields in the phase diagrams,
are qualitatively consistent with the experiments in PrV$_2$Al$_{20}$.
