A lot of people are unclear about quarantine rules. The short version is that things are much easier now and, generally, your dog does not need to be quarantined. Within the EU, pets are issued passports by a veterinarian. In the US, they are issued travel documents, but not an actual passport. You should allow 6 months to complete the process in the first instance, but this can be shorted. For instance, my vet gave Fritz a one-dose rabies vaccination instead of a two-dose vaccination that would have been spread over two months. This shortened the process and he was able to make an international trip at about four and a half months old.
The process does change regularly, so you should check with the country’s department of agriculture and/or border agency before you book your travel. You can find more information here.