I'm not sure if this will be accurate at all or not, but perhaps call your vet and ask if they put a tube down your pups throat. I know the first time Clyde went under anesthesia they put a tube down his throat and he had a horrible cough for two days afterwards because of all the irritation from the tube. I don't know much about vet procedure, but I assume that this is common practice?
Or maybe your dog picked something up at the vets office? It is really hard to tell, but I would go ahead and call my vet and ask his or her advice as to what you should do.
Oh! And by the way
Welcome to YT!!!! I hope that you find an answer as to why your pup is coughing! Good luck