The author studies several \(q\)-analogues of a family of identities involving Stirling numbers of the first and second kind. In the derivation of these identities a central role is played by the Arik-Coon-type \(q\)-boson operators obtained in [M. Arik and D. D. Coon, J. Math. Phys. 17, 524-526 (1976; Zbl 0941.81549)].