Generalized linear models are a standard statistical tool for modeling the relation between a response variable and a set of covariates. Despite their popularity, they can incur in misspecification problems that could negatively impact inferential conclusions. A semi-parametric solution adopted in frequentist literature uses the quasi-likelihood function, which relies on the second-order assumptions, in place of the usual likelihood. This approach yields increased flexibility since only the first two moments of the data generator are specified rather than its entire distribution. We propose to integrate this solution in the Bayesian paradigm introducing the quasi-posterior distribution. This quantity represents a coherent Bayesian update according to the generalized Bayes notion. We show that quasi-posterior approximates the regression coarsened posterior in the case of exponential families, providing new insights on the choice of the coarsening parameter. Asymptotically, the quasi-posterior converges in total variation to a normal distribution, has important connections with the loss-likelihood bootstrap posterior, and is also well-calibrated in terms of frequentist coverage. Moreover, the loss-scale parameter has a clear interpretation in terms of the dispersion parameter, leading to the consolidated method of moments estimator for its quantification. We provide some applications to overdispersed counts and heteroscedastic continuous data.