Using first-principles calculations, we have studied the electronic structure and magnetic properties in a IIIA-nitride monolayer doped with Cu. The substitutional Cu impurity induces a global magnetic moment of 2.00 μ_B. All the doped systems are half metallic at the GGA level, but are magnetic semiconductors at the HSE06 level. As the atomic number of the IIIA elements increases, the increasing covalency leads to a tendency towards delocalization of the local magnetic moment. The ground state magnetism is determined by the competition between the ferromagnetic p–p/p–d hybridization interaction and the anti-ferromagnetic super-exchange interaction. By using external strain, the magnetic ground state can be deliberately tuned.