  We show that the pair-density-wave (PDW) superconducting state emergent in
extended Heisenberg-Hubbard models in two-leg ladders is topological in the
presence of an Ising spin symmetry and supports a Majorana zero mode (MZM) at
an open boundary and at a junction with a uniform $d$-wave one-dimensional
superconductor. Similarly to a conventional finite-momentum paired state, the
order parameter of the PDW state is a charge-$2e$ field with finite momentum.
However, the order parameter here is a {\it quartic} electron operator and
conventional mean-field theory cannot be applied to study this state. We use
bosonization to show that the 1D PDW state has a MZM at a boundary. This
superconducting state is an exotic topological phase supporting Majorana
fermions with finite-momentum pairing fields and charge-$4e$ superconductivity.
