Algorithms of implicit discretization for generalized homogeneous systems having a possible discontinuity only at the origin are developed. They are based on the transformation of the original system to an equivalent one which admits an implicit or a semi-implicit discretization scheme preserving the stability properties of the continuous-time system. Namely, the discretized model remains finite-time stable (in the case of negative homogeneity degree) and practically fixed-time stable (in the case of positive homogeneity degree). The theoretical results are supported with numerical examples.