As the coefficients table suggests (
z value), the GLM model treats the parameter estimates (scaled by their standard errors) as following (based on asymptotic distribution) a standard normal or Z distribution. The t-distribution only applies when we assume (some function of) the data follow a normal distribution, too. So in linear regression, when the outcome is assumed to have a normal distribution, the parameters then follow a t distribution. But for (non-linear) GLMs this is explicitly not assumed.
In brief, then, the t distribution doesn't really enter the picture.
The Chi-squared distribution can be used to test differences between models as a whole, for example to compare the 'null' model and its deviance to your fitted model with its residual deviance. here are some more details about that test and its rationale.
The F distribution is simply the distribution of the ratio of two (independent) Chi-squared random variables but it doesn't necessarily have an obvious application in evaluating your model.
It's just as important to look at your data and try to check or test the model assumptions - a standard move for Poisson GLM is to also fit a negative binomial and compare the fits. Consider too whether any of your predictors have interactions or are highly correlated.