We aim to present a innovation extension of the mean-square stochastic integral of a process  on time scale based on a $\mathfrak{q}$-operator and fractional $\mathfrak{q}$-operator. By presenting the $\mathfrak{q}$-convex definition for a stochastic process and using quantum operators, we were able to prove some well-known inequalities such as the Hermit-Hadamardand the Jensen for the processes.