In this paper, by using the techniques of the q-exponential generating series, we extend a well-known two-parameter identity for the Appell polynomials to the q-Appell polynomials of type I and II. More precisely, we obtain two different q-analogues of such an identity. Then, we specialize these identities for some q-polynomials arising in combinatorics, in q-calculus or in the theory of orthogonal polynomials. In particular, we consider the generalized q-Bernoulli and q-Euler polynomials and then we deduce some further identities involving the Bernoulli and Euler numbers. In this way, as a byproduct, we derive the symmetric Carlitz identity for the Bernoulli numbers. Finally, we find a (non-symmetric) q-analogue of Carlitz's identity involving the q-Bernoulli numbers of type I and II.