T-B₃N₃ is the only allotrope of Boron Nitride that has metallic properties and Since the elements of this composition, nitrogen and boron, are light elements, as well as the bonding between them in this compound, are hard, these two properties along with the metallic properties, potentially make this composition as a good candidate for phonon-mediated superconductivity with a suitable superconducting transition temperature. In this thesis, relaxation calculations were initially performed to obtain the precise position of the atoms in the T-B₃N₃material. Then, based on the resulted relaxed crystal, electronic structure calculations such as band structure calculations, electronic density of states, and projected density of states are performed. As follows, phonon structure calculations and Electron-Phonon Coupling (EPC) are performed. Since pressure always can be a good factor for increasing the density of states at the Fermi level, the tensile strength of the bindings, and consequently, increasing superconductivity transition temperature, in this research, the effect of both positive and negative pressure on T-B₃N₃ compound has been investigated. Also, according to the projected density of states calculations in relaxed and under-pressure conditions, the contribution of the orbitals near the Fermi level has been investigated. Based on assuming the LDA and two different GGAs pseudopotentials, we performed relaxation and electronic calculations. It has been shown that these results are almost independent of the choosing the pseudopotential types. Accordingly, all following self-consistent calculations, electronic and phonon structures and electron-phonon coupling are done using the pseudopotential LDA function. To do this, all calculations are performed based on the density functional theory and using Quantum Espresso software.