We construct quadratic stochastic processes (QSP) (also known as Markov processes of cubic matrices) in continuous and discrete times. These are dynamical systems given by (a fixed type, called $σ$) stochastic cubic matrices satisfying an analogue of Kolmogorov-Chapman equation (KCE) with respect to a fixed multiplications (called $μ$) between cubic matrices. The existence of a stochastic (at each time) solution to the KCE provides the existence of a QSP called a QSP of type $(σ| μ)$. In this paper, our aim is to construct and study trajectories of QSPs for specially chosen notions of stochastic cubic matrices and a wide class of multiplications of such matrices (known as Maksimov's multiplications).