We describe an implementation of ab-initio methodology to compute inelastic shot noise signals due to electron-vibration scattering in nanoscale junctions. The method is based on the framework of non-equilibrium Keldysh Green's functions with a description of electronic structure and nuclear vibrations from density functional theory. Our implementation is illustrated with simulations of electron transport in Au and Pt atomic point contacts. We show that the computed shot noise characteristics of the Au contacts can be understood in terms of a simple two-site tight-binding model representing the two apex atoms of the vibrating nano-junction. We also show that the shot noise characteristics of Pt contacts exhibit more complex features associated with inelastic interchannel scattering. These inelastic noise features are shown to provide additional information about the electron-phonon coupling and the multichannel structure of Pt contacts than what is readily derived from the corresponding conductance characteristics. We finally analyze a set of Au atomic chains of different lengths and strain conditions and provide a quantitative comparison with the recent shot noise experiments reported by Kumar et al. [Phys. Rev. Lett. 108, 146602 (2012)].