It is commonly observed in medical and financial studies that large volume of time series of count data are collected for several variates. The modelling of such time series and the estimation of parameters under such processes are rather challenging since these high dimensional time series are influenced by time-varying covariates that eventually render the data non-stationary. This paper considers the modelling of a bivariate integer-valued autoregressive (BINAR(1)) process where the innovation terms are distributed under non- stationary Poisson moments. Since the full and conditional likelihood approaches are cumbersome in this situation, a Generalized Quasi-likelihood (GQL) approach is proposed to estimate the regression effects while the serial and time-dependent cross correlation effects are handled by method of moments. This new technique is assessed over several simulation experiments and the results demonstrate that GQL yields consistent estimates and is computationally stable since few non-convergent simulations are reported.