Quantitative phase imaging (QPI) is a label-free technique that provides optical path length information for transparent specimens, finding utility in biology, materials science, and engineering. Here, we present QPI of a three-dimensional (3D) stack of phase-only objects using a wavelength-multiplexed diffractive optical processor. Utilizing multiple spatially engineered diffractive layers trained through deep learning, this diffractive processor can transform the phase distributions of multiple two-dimensional objects at various axial positions into intensity patterns, each encoded at a unique wavelength channel. These wavelength-multiplexed patterns are projected onto a single field of view at the output plane of the diffractive processor, enabling the capture of quantitative phase distributions of input objects located at different axial planes using an intensity-only image sensor. Based on numerical simulations, we show that our diffractive processor could simultaneously achieve all-optical QPI across several distinct axial planes at the input by scanning the illumination wavelength. A proof-of-concept experiment with a 3D-fabricated diffractive processor further validates our approach, showcasing successful imaging of two distinct phase objects at different axial positions by scanning the illumination wavelength in the terahertz spectrum. Diffractive network-based multiplane QPI designs can open up new avenues for compact on-chip phase imaging and sensing devices.