The Monte Carlo Nucleon Transport (MONC) code for nucleon transport is extended for below 20-MeV proton transport using the ENDF and EXFOR databases. It is used to simulate the p + 7Li reaction up to 20-MeV proton energies, with the produced neutron spectra reported here. The simulated results are compared with the calculated values from other available codes like PINO, EPEN, and SimLiT, as well as experimental data. The spectra reported here can be used to get the neutron cross section for the quasi-monoenergetic neutron reaction and will help to subtract the low-energy contribution. The primary neutron spectra and its transport are useful, as this reaction has the potential for accelerator-based boron neutron capture therapy. The backing materials are used to fully stop the proton beam, hence the contributions of the neutrons from backing materials are estimated. It is found that tantalum is a good backing material below ~8 MeV and that carbon is better at higher energies.