Putting aside the breathing issues that are normal and daily occurrences, the Pug dog will be vulnerable to some respiratory medical issues.
Stenotic Nares -
This is not uncommon with brachycephalic, short-nosed dogs. This is a physical condition in which the dog's nostril are too narrow to allow for proper breathing.
This is also referred to as pinched nostrils. It is a congenital trait, which means that it is passed down genetically, however it cannot be bred out of the Pug due to facial structure that gives the Pug his unique appearance.
For some, this can be noticed at birth, however many veterinarians recommend waiting on surgery because some Pugs outgrow this.
Additionally, for some puppies, the nostrils may close off more during the teething phase but re-open enough for easy breathing once the pup's adult teeth have grown in.
Due to the changes that can occur during puppy-hood, surgery is usually only recommended for severe cases with Pug puppies
The most common and noticeable sign that this is a problem is if the dog can breathe just fine through his mouth, but not his nose. When the Pug's mouth is open, breathing will be normal; when closed, he will gasp, grunt and struggle for air.