In this paper, a hybrid Lagrangian-Eulerian topology optimization (LETO) method is proposed to solve the elastic force equilibrium with the Material Point Method (MPM). LETO transfers density information from freely movable Lagrangian carrier particles to a fixed set of Eulerian quadrature points. The transfer is based on a smooth radial kernel involved in the compliance objective to avoid the artificial checkerboard pattern. The quadrature points act as MPM particles embedded in a lower-resolution grid and enable a sub-cell multi-density resolution of intricate structures with a reduced computational cost. A quadrature-level connectivity graph-based method is adopted to avoid the artificial QR patterns commonly existing in multi-resolution topology optimization methods. Numerical experiments are provided to demonstrate the efficacy of the proposed approach.