While you can estimate distances under a handful of models in R, to my knowledge you are only able to estimate trees under the GTR model when using likelihood-based approaches (as, for example, in phangorn's optim.pml() and others). Unfortunately, I think you'll have to do this outside of R. Most sophisticated phylogenetic estimation isn't done in R, anyway.

I don't have time to skim through the PAUP manual and provide detailed instructionsl, but you can calculate distances under the GTR model and estimate a tree generally following this tutorial.