You should be cautious in trying to find an $R^2$-like measure for modeling of a discrete response, as a generalized linear model of a discrete response is based on maximum-likelihood estimates rather than least squares. In R, the
summary of a
glm model for discrete responses reports the deviance and the Akaike Information Criterion (AIC). This UCLA web page shows how to use a test on residual deviance to evaluate the overall goodness of fit of a Poisson regression. The AIC may also be useful in comparing models.
There are several types of pseudo-$R^2$ that have been proposed to be comparable to the $R^2$ values generated in least-squares analyses. These pose issues of which you should be aware, as explained nicely by this answer from @Gung and the links therein. As noted in that answer, $R^2$ can be a slippery concept even in least squares.