The realization of the spin-orbit coupling (SOC) driven j_eff = 1/2 anti-ferromagnetic (AFM) Mott insulating state in Sr₂IrO₄ has generated a huge interest in the field of 5d transition metal oxides. Recently, much work has been done on the electron and hole doped materials, particularly because doping can reveal many properties of the underlying structure. In this context, we have studied a minimal model, an extended (t-t'-t'') Hubbard model on a square lattice (in terms of which the upper and lower Hubbard bands can be described) using the Hartree-Fock mean field theory to understand the quantum many-body instabilities of the AFM state in the doped relativistic Mott insulator Sr₂IrO₄. Our results for the relevant parameters required to describe the material show that the AFM state is stable for a range of both electron and hole doping concentration. Interestingly, a mixed phase of metallic and insulating regions leading to percolative conduction occurs in case of hole doping. Our analysis of the stability of the AFM state in the doped extended Hubbard model of the present work, with respect to the transverse perturbations shows that the sign of the third nearest neighbor interaction plays a crucial role in determining the stability of the AFM state away from half-filling.