Due to their ability to capture complex interactions between random variables, copula models are gaining increasing attention. When it comes to bivariate data modeling, one important area of statistical theory is the construction of families of distributions with specified marginals. An FGM-QEXD (bivariate quadratic exponential Farlie Gumbel Morgenstern distribution) is derived from the FGM copula and the new quadratic exponential marginal distribution, and is inspired by this. The statistical features of the FGM-QEXD are studied, encompassing: the conditional distribution, regression function, moment generating function, and correlation coefficient. Additionally, reliability measures were obtained, including the survival function, hazard rate function, mean residual life function, and vitality function. The model parameters are estimated via maximum likelihood (ML) and Bayesian methodologies. Furthermore, asymptotic confidence ranges for the model parameter are obtained. Monte Carlo simulation analysis is employed to evaluate the efficacy of both ML and Bayesian estimators. Two real-world datasets are employed to prove that FGM-QEXD is more flexible than the bivariate Weibull Farlie–Gumbel–Morgernstern (FGM), bivariate Lomax FGM, bivariate inverse Lomax FGM, bivariate Rayleigh FGM, bivariate Burr XII FGM, and bivariate Chen FGM distributions.