Recently, room-temperature superconductivity has been reported in a nitrogen-doped lutetium hydride at near-ambient pressure [Dasenbrock-Gammon et al., Nature 615, 244 (2023)]. The superconducting properties might arise from Fm$\bar{3}$m-LuH_3-δN_ε. Here, we systematically study the phase diagram of Lu–N–H at 1 GPa using first-principles calculations, and we do not find any thermodynamically stable ternary compounds. In addition, we calculate the dynamic stability and superconducting properties of N-doped Fm$\bar{3}$m-LuH₃ using the virtual crystal approximation (VCA) and the supercell method. The R3m-Lu₂H₅N predicted using the supercell method could be dynamically stable at 50 GPa, with a T_c of 27 K. According to the VCA method, the highest T_c is 22 K, obtained with 1% N-doping at 30 GPa. Moreover, the doping of nitrogen atoms into Fm$\bar{3}$m-LuH₃ slightly enhances T_c, but raises the dynamically stable pressure. Our theoretical results show that the T_c values of N-doped LuH₃ estimated using the Allen–Dynes-modified McMillan equation are much lower than room temperature.