In this paper, we establish certain combinatorial interpretation for $q$-analogue of $r$-Whitney numbers of the second kind defined by Corcino and Ca\~{n}ete in the context of $A$-tableaux. We derive convolution-type identities by making use of the combinatorics of $A$-tableaux. Finally, we define a $q$-analogue of $r$-Dowling numbers and obtain some necessary properties including its Hankel transform.

A q-analogue of r-Whitney numbers of the second kind, denoted by Wm,r[n, k]q, is defined by means of a triangular recurrence relation. In this paper, several fundamental properties for the q-analogue are established including other forms of recurrence relations, explicit formulas and generating functions. Moreover, a kind of Hankel transform for Wm,r[n, k]q is obtained.