Thermoelectric (TE) materials are semiconductors that can directly convert between the flow of thermal energy and the flow of electrical energy. This technology can be used for generation of electricity and for heating or refrigeration. Application of this technology includes recovering energy from waste heat, powering satellite systems in space, and providing thermal management. In this study, a second-order accurate hybrid implicit-explicit finite-volume method algorithm and computer code was developed to analyze the energy and the electric current flow equations that govern the three-dimensional temperature and electric potential distributions in a TE couple with temperature dependent material properties. The TE couple consists of the p and n TE legs with square cross sections that are connected by nickel plates to conduct electricity and are clad by thermally conducting but electrically non-conducting plates that separate the TE material from the hot and the cold temperature sources. For this TE couple, the region between the p and n TE legs can be filled with an inert gas or an insulating material. In the analysis, the thermal and electrical contact resistances at junctions where dissimilar materials meet are accounted for. For this TE couple, effects of the following parameters were investigated: (1) specified temperature at the hot and cold sources (2) insulation material with thermal conductivities of 1, 0.1, 0.01, and 0.001 [W over m·K], (3) inert gas with effective heat transfer coefficients of 0, 100, and 200 [W over m²·K], (4) thermal and electrical contact resistances, and (5) imposed electric potential difference across the TE couple. A grid sensitivity study was performed to ensure that the results obtained are grid independent.