Similar to most semiconductors, low-dimensional GaN materials also have the problem of asymmetric doping, that is, it is quite difficult to form p-type conductivity compared to n-type conductivity. Here, we have discussed the geometry, structure, and electronic defect properties of a two-dimensional graphene-like gallium nitride (g-GaN) monolayer belonging to the group III-V compounds, doped with different elements (In, Mg, Zn) at the Ga site. Based on first principles calculations, we found that substituting Ga (low concentration impurities) with Mg would be a better choice for fabricating a p-type doping semiconductor under N-rich conditions, which is essential for understanding the properties of impurity defects and intrinsic defects in the g-GaN monolayer (using the "transfer to real state" model). Moreover, the g-GaN monolayer is dynamically stable and can remain stable even in high-temperature conditions. This research provides insight for increasing the hole concentration and preparing potential high-performance optoelectronic devices using low-dimensional GaN materials.