This paper is devoted to study the numerical solution of a system of Quadratic integral equations (SQIEs) in position and time in the space using brand-new orthogonal polynomials described as eighth-kind Chebyshev polynomials (CP8K). Specific kinds of Gegenbauer polynomials are these polynomials. By using the separation of variables technique the SQIEs in position and time transformed to SQIEs in position, then by applying CP8K, consequently, a system of linear algebraic equations (SLAEs) is constructed. The Banach fixed point theorem is employed to demonstrate the existence and uniqueness of the SQIEs solution. Additionally, the solution’s convergence and stability are studied. Some numerical examples are constructed to illustrate the efficiency and applicability of the method. All the computational are obtained by Maple 18 software. Finally, computer simulations can be obtained to demonstrate the mathematical results.