The main purpose of this paper is to define a new extension of Hermite-Hermite matrix polynomials (HHMP) and derive some properties such as recurrence relations and matrix differential equation for generalized Hermite-Hermite matrix polynomials. We derive addition and multiplication theorem, and summation formula for generalized HHMP. Furthermore, via integral transform, the new families of Chebyshev-Chebyshev, Legendre-Legendre, Chebyshev-Hermite and Legendre-Hermite matrix polynomials are introduced, from which a variety of
interesting results follows as special cases.