The Lennard-Jones potential is perhaps one of the most widely-used models for the interaction of uncharged particles, such as noble gas solids. The phase diagram of the classical LJ solid is known to exhibit transitions between hcp and fcc phases. However, the phase behaviour of the quantum Lennard Jones solid remains unknown. Thermodynamic integration based on path integral molecular dynamics and lattice dynamics calculations are used to study the phase stability of the hcp and fcc Lennard Jones solids. The hcp phase is shown to be stabilized by quantum effects in PIMD while fcc is shown to be favoured by lattice dynamics, which suggests a possible re-entrant low pressure hcp phase for highly quantum systems. Implications for the phase stability of noble gas solids are discussed. For parameters equating to Helium, the expansion due to zero-point vibrations is associated with quantum melting: neither crystal structure is stable at zero pressure. This dataset contains input files and run scripts for the quasiharmonic calculations using GULP, and the PIMD calculations using i-PI and LAMMPS. See README.txt files within each archive for specific details on how to run and analyze the calculations.