The only reason why a vet should cut into the quick is if the nails are too long. If you let your dogs nails get long and go a good while without cutting them the quick grows which isn't good for the dog. Other than that I don't know why they would purposfully cut the quick. I had a dog that did that once except it was the first time we ever cut his nails when he was a puppy, surprisingly one of his toenails were hollow... no quick... yea that was weird. We didn't know that until it got infected. We woke up a few mornings after he had his first nail trim and his toe was as big as half a golf ball! We took him to the vet got him some antibiotics, wrapped his foot and regularly put Nolvasan (an antiseptic ointment) on his foot. Also he had to where a cone on his head to keep him from tearing off the bandage. It took about 4 or 5 days to heal good enough we could take the cone off his head and remove the bandage. I don't remember exactly how long it took to heal completely though. Hope this helps.
What antibiotics did the vet prescribe?
And if your dogs quick under the nails weren't too long I think the vet should pay to have the infection treated.
I can't imagine why it was even necessary to cut into the quick. But, the damage has been done so you can only deal with what's happening now. Cellutitis is very dangerous, and I wonder if your dog should be an in-patient on IV antibiotics. After 4 weeks with no change, you really need to decide whether or not staying with this vet is the best thing for your dog. If you have a large teaching university close to you, give their department of veterinary medicine a call and explain the situation. It sounds like your dog needs a much higher standard of care.
If you haven't posted there already, you can get a veterinarian's opinion for free here at Med Help. Dr. Cheng generally answers questions daily, and you can access that forum by clicking the link labeled Ask a Vet (Pet Health) on the lower right side of this screen. If it were me, I'd be getting on the phone to check for second opinions.