The pseudogap, d-wave superconductivity and electron-boson coupling are three intertwined key ingredients in the phase diagram of the cuprates. Sr$_2$IrO$_4$ is a 5d-electron counterpart of the cuprates in which both the pseudogap and a d-wave instability have been observed. Here, we report spectroscopic evidence for the presence of the third key player in electron-doped Sr$_2$IrO$_4$: electron-boson coupling. A kink in nodal dispersion is observed with an energy scale of ~50 meV. The strength of the kink changes with doping, but the energy scale remains the same. These results provide the first noncuprate platform for exploring the relationship between the pseudogap, d-wave instability and electron-boson coupling in doped Mott insulators.