We establish new upper bounds for the Berezin number and Berezin norm of operator matrices, which are refinements of existing bounds. Among other bounds, we prove that if (Formula presented.) is an (Formula presented.) operator matrix with (Formula presented.) for (Formula presented.), then (Formula presented.) and (Formula presented.) where (Formula presented.) (Formula presented.) if ij. We also provide examples which illustrate these bounds for some concrete operators acting on the Hardy-Hilbert space.