  A pair-density-wave (PDW) is a novel superconducting state with an
oscillating order parameter. A microscopic mechanism that can give rise to it
has been long sought but has not yet been established by any controlled
calculation. Here we report a density-matrix renormalization group (DMRG) study
of an effective $t$-$J$-$V$ model, which is equivalent to the Holstein-Hubbard
model in a strong-coupling limit, on long two-, four- and six-leg triangular
cylinders. While a state with long-range PDW order is precluded in one
dimension, we find strong quasi-long-range PDW order with a divergent PDW
susceptibility as well as the spontaneous breaking of time-reversal and
inversion symmetries. Despite the strong interactions, the underlying Fermi
surfaces and electron pockets around the $K$ and $K^\prime$ points in the
Brillouin zone can be identified. We conclude that the state is
valley-polarized and that the PDW arises from intra-pocket pairing with an
incommensurate center of mass momentum. In the two-leg case, the exponential
decay of spin correlations and the measured central charge $c\approx 1$ are
consistent with an unusual realization of a Luther-Emery liquid.
