I have a Poisson GLM in R that is overdispersed, so I fit a quasipoisson GLM, however the residual deviance nor the degree of freedom change. Can that happen? What does it mean in that case?
Yes, it can and is perfectly fine. Actually in this case, the estimated coefficients should be the same as well as the residual deviance and the degrees of freedom! But have a look at the standard errors of the estimated coefficients. You will see larger standard errors when fitting a quasipoisson GLM. It means that you are fitting a more flexible model that allows an over-dispersion. Therefore, you have larger confidence intervals.