The authors deal with matricial versions of generalized Stieltjes type moment problems. In fact they study two slightly different versions of truncated generalized matricial moment problems of Stieltjes type, constructing distinguished solutions for both of them. The approach is mainly algebraic, based on the use of some particular matrix polynomials, and the solutions are nonnegative Hermitian Borel measures on the real axis, concentrated on a finite number of points.