The relic abundance of asymmetric Dark Matter particles in the scalar-tensor model is analyzed in this article. We extend the numeric and analytic calculation of the relic density of the asymmetric Dark Matter in the standard cosmological scenario to the nonstandard cosmological scenario. We focus on the scalar-tensor model. Hubble expansion rate is changed in the nonstandard cosmological scenario. This leaves its imprint on the relic density of Dark Matter particles. In this article we investigate to what extent the asymmetric Dark Matter particle's relic density is changed in the scalar-tensor model.We use the observed present day Dark Matter abundance to find the constraints on the parameter space in this model. *