The pair density wave (PDW) is an unconventional superconducting state that provides a unified description of intertwined orders in quantum materials. However, the microscopic mechanism of PDW in strongly-correlated electron systems remain a subject of intense theoretical and experimental interest. The recent discovery of high-temperature superconductivity in the multilayer nickelates La3Ni2O7 and La4Ni3O10 indicates that the interlyaer antiferromagnetic exchange J⊥ is a key pairng driver. In this work, we employ degenerate perturbation theory and the density-matrix renormalization group (DMRG) simulations to investigate a trilayer fermion t-J model characterized by strong interlayer antiferromagnetic exchange coupling J⊥. At the filling of two electrons per unit cell, the system forms a charge-gapped insulator composed of interlayer spin-singlet Cooper pairs carrying a layer pseudospin 1/2 . Upon doping this insulator, the low-energy physics of these mobile Cooper pairs is effectively mapped onto a two-species hard-core boson t-J-V model. Through DMRG simulations of both the fermion and effective boson models, we observe both the single boson condensation at non-zero momentum and the finite momentum interlyer fermion singlet pairing, signaling a robust PDW state. We further identify an interlayer coherent state characterized by antiferromagnetic bosonic order. Our results establish a novel pathway for achieving PDW states in strongly correlated systems under strong interlayer exchange J⊥ in the multilayer system.