The bivariate generalized Poisson distribution has been found useful in describing correlated count data. In this paper, a bivariate generalized Poisson regression model based on copula functions is defined and studied. The marginal means of the bivariate model are functions of the explanatory variables. The parameters of the regression model are estimated by using the maximum likelihood method. Some test statistics including goodness of fit are discussed. One numerical data set that exhibits over-dispersion is used to illustrate the techniques. Effect of the correlation on the biases and standard errors of the parameter estimates is studied through simulation.