Generalized autoregressive linear models with moving average (GARMA) allow the modeling of discrete time series with correlation structure similar to those of ARMA’s models. In this work we developed an extension of a univariate Poisson GARMA model by considerating the specification of a bivariate Poisson model through the distribution presented on Kocherlakota and Kocherlakota (1992), which will be called Poisson BGARMA model. The proposed model not only is suitable for stationary discrete series, but also allows us to take into consideration the effect of seasonality and trend. The investigation of the usual properties of the maximum likelihood estimators (bias, efficiency and distribution) was performed using Monte Carlo simulations. Aiming to compare the performance and compliance of the proposed model, it was applied to two pairs of series of bivariate count data. The first pair is the monthly counts of neonatal deaths to two lanes of days. The second pair refers to counts of daily car accidents in two distinct periods: afternoon and evening. The results of our model when compared with those obtained by fitting a bivariate Vector Autoregressive Gaussian model (VAR) indicates that the Poisson BGARMA model is able to proper capture the variability of bivariate vectors of real time series of count data, producing forecasts with acceptable errors and allowing one to obtain probability forecasts.