In this paper, we derive several combinatorial identities involving the q-derangement numbers (for the major index) and many other q-numbers and q-polynomials of combinatorial interest, such as the q-binomial coefficients, the q-Stirling numbers, the q-Bell numbers, the q-Pochhammer symbol, the Gaussian polynomials, the Rogers-Szegő polynomials and the Galois numbers, and the Al-Salam-Carlitz polynomials. We also obtain two determinantal identities expressing the q-derangement numbers as tridiagonal determinants and as Hessenberg determinants.