In this work, we study the mass spectrum of the $Omega_{ccc}$ and $Omega_{bbb}$ baryons up to the $N=2$ shell within a nonrelativistic constituent quark model (NRCQM). The model parameters are adopted from the determinations by fitting the charmonium and bottomonium spectra in our previous works. The masses of the $Omega_{ccc}$ and $Omega_{bbb}$ baryon states predicted in present work reasonably agree with the results obtained with the Lattice QCD calculations. Furthermore, to provide more knowledge of the $Omega_{ccc}$ and $Omega_{bbb}$ states, we evaluate their radiative decays with the available masses and wave functions from the potential model.