Fluoride-salt-cooled high-temperature reactor (FHR) designs use FLiBe salt as a coolant, and thus generate a significant amount of tritium, which becomes an operational challenge in terms of radioactive protection. In the design of a tritium management program, it is important to accurately estimate the spatial and temporal tritium production rate. This is the scope of this work, achieved by three-dimensional neutronics simulations throughout all areas of an FHR, including the FLiBe coolant, graphite moderator, graphite reflector, and fuel planks with embedded tristructural-isotropic (TRISO) particles. Note that the subsequent tritium transport within the system, and the methods for its removal are not addressed. The lithium isotope 6Li in the coolant is expected to be the main tritium precursor and thus the primary focus. Unlike the TRISO fuel particles that contain tritium by design, the FLiBe coolant directly interacts with the primary heat exchangers, which provides a large avenue for tritium to escape the primary loop. This work seeks to ensure that not only are the models accurate but also the data libraries and physics involved with calculating tritium production. A computational methodology using the Monte Carlo N-Particle MCNP6.1 and Standardized Computer Analyses for Licensing Evaluation SCALE6.2 code packages has been developed and is presented in the paper. Impact of the fuel design, as well as modeling approximations have been examined. Initial tritium production rates have a strong dependence on fuel enrichment (varying between roughly 13,700 and 9,300 Ci/day), while equilibrium rates converge to the same levels of roughly 3,800 Ci/day. For the reference case, it was found that 21.5 MCi of tritium are produced by the end of a 100-year reactor lifetime.