The magnetism of two-dimensional material is an important research topic. In particular, the long-range magnetic order of two-dimensional material is of great significance in theoretical research and practical application. According to the Mermin-Wagner theory, the isotropic Heisenberg model in a two-dimensional system cannot produce long-range magnetic orders at non-vanishing temperatures. Considering the existence of strong magnetic anisotropy, possible two-dimensional long-range magnetic orders may exist in 5d atom doped two-dimensional aluminum nitride (AlN) monolayer. This research is performed by first-principles calculations based on the density functional theory. Geometries, electronic structures, magnetic properties, and magnetic anisotropy energies from spin-orbital coupling effects in AlN monolayers doped by 5d transition metal atoms (Hf, Ta, W, Re, Os, Ir, Pt, Au, and Hg) are calculated. Four kinds of supercells are used in the calculation, i.e, 22, 33, 44, and 55, with one aluminum atom substituted by one 5d atom. Projection augmented wave method is used to describe the interaction between the valence electrons and the ions. The plane wave is used to expand the wave function of the valence electron. For an optimized geometry, the bond length between the 5d metal atom and the nearest N atom is the largest in Hg-doped supercells, which is 2.093 , followed by the Au, Hf, Pt, Ta, and Ir according to the order of bond length magnitude. For the densities of states (DOSs), obvious impurity energy levels appear in the forbidden bands. For all the supercells, spin-up and spin-down DOSs of Ta and Ir doped systems are symmetric, indicating non-magnetic states. DOSs of Hf, W, Re, and Os doped systems are asymmetric, indicating magnetic states. For Pt, Au, and Hg, DOSs are symmetric in 22 supercells, but asymmetric in the 33, 44, and 55 supercells. Total magnetic moments and the spin densities are also given. In 55 supercells, they are 1.00, 0.00, 0.39, 1.99, 1.17, 0.00, 1.00, 2.00, and 1.00 for Hf, Ta, W, Re, Os, Ir, Pt, Au, and Hg, respectively. The magnetic moment is mainly concentrated in the vicinity of the 5d atoms. The energy differences between ferromagnetic and antiferromagnetic states are calculated. For Hf, Re, Pt and Au systems, the differences in 48 supercells reach the maximum values of -187.2563 meV, 286.2320 meV, -48.0637 meV and -61.7889 meV, respectively. The results indicate that there is a strong interaction between the magnetic centers. Magnetic anisotropy energy originating from spin-orbital effect is calculated in the 44 supercells. For the Re system, it is the highest, reaching 11.622 meV. For W, Os, and Au, the values are larger than 1 meV, showing strong magnetic anisotropies. The magnetic anisotropy can produce a spin wave energy gap, resulting in long-range magnetic orders. Based on the results above, it is predicted that with appropriate 5d atoms and suitable doping concentration, two-dimensional long-range magnetic orders may exist in 5d transition metal atom doped AlN monolayers.