Given a pair of Hilbert space linear operators \((V_1,V_2)\) and a complex number q, \((V_1, V_2)\) is said to be q-commuting if \(V_1 V_2 = q V_2 V_1\) . This amounts to a generalization (or deformation) of the case \(q=1\) corresponding to a commuting pair of operators. Here we consider the case where \(V_1\) and \(V_2\) are both isometries and q is unimodular complex number. Given a data set \(\Xi = ({\mathcal F}, {\mathcal K}_u; P,U, W_1, W_2)\) consisting of (i) Hilbert spaces \({\mathcal F}\) , \({\mathcal K}_u\) , (ii) a projection operator P and unitary operator U acting on \({\mathcal F}\) , and (iii) a q-commuting unitary operator pair \((W_1, W_2)\) acting on \({\mathcal K}_u\) , we associate a functional-model q-commuting isometric pair \((V_{\Xi ,1}^{(q)}, V_{\Xi ,2}^{(q)})\) acting on the Hilbert space \(H^2({\mathcal F}) \oplus {\mathcal K}_u\) . Conversely, given any q-commuting isometric pair \((V_1, V_2)\) , we show that there is a choice of data set \(\Xi \) so that \((V_1, V_2)\) is jointly unitarily equivalent to \((V_{\Xi ,1}^{(q)}, V_{\Xi ,2}^{(q)})\) . Furthermore, the data set \(\Xi \) is uniquely determined by \((V_1, V_2)\) up to a unitary change of coordinates in the spaces \({\mathcal F}\) and \({\mathcal K}_u\) . This model amounts to an explicit continuous q-deformation ( \(q \mapsto (V_{\Xi ,1}^{(q)}, V_{\Xi ,2}^{(q)})\) for \(q \in {\mathbb T}\) ) of the model of Berger-Coburn-Lebow for the \(q=1\) case. A q-commuting pair \((V_1,V_2)\) is said to be doubly q-commuting, if in addition, the relation \(V_2V_1^*=qV_1^*V_2\) holds. We characterize in terms of the data set \(\Xi \) when it is the case that \((V_{\Xi ,1}^{(q)}, V_{\Xi ,2}^{(q)})\) is actually doubly q-commuting. It turns out that \((V^{(q)}_{\Xi ,1}, V^{(q)}_{\Xi ,2})\) is doubly q-commuting for one q if and only if it is doubly q-commuting for all q. For the case where \((V_1, V_2)\) is a doubly q-commuting pair of shift operators, we show how to model \((V_1, V_2)\) as a continuous q-deformation of the coordinate multipliers \((M_{z_1}, M_{z_2})\) on a vectorial Hardy space \(H^2_{{\mathbb D}^2} \otimes {\mathcal F}\) of the bidisk, thereby extending a result of Słociński for the case \(q=1\) . We also obtain Wold decompositions for a pair of q-commuting as well as doubly q-commuting isometries, obtained earlier by Popovici and Bercovici-Douglas-Foias for the commuting case and by Słociński for the doubly commuting case.