Key: 1 or 2 eyes? Allergy of the eyes are almost always both eyes (allergic conjunctivitis). So if, in fact, only one is affected, it is more likely that something got in that eye, you rubbed it and became inflamed, or a bacteria infected it (bacterial conjunctivitis); could also be a virus-also both eyes. If you sneeze or cough a lot, a little vessel may break & cause bloodshot eye (not serious). Could be dog hair.
Allergies. It sounds like you have an allergy to your new dog. It is fairly common to have allergies affect your eyes without an affect on your nose and sinuses, and visa versa. But don't get rid of your puppy just yet. There are a number of ways to help with your symptoms including eye drops plus or minus allergy pills.
Possibly. Animal dander is a very common allergen. If your eyes are watering/itching around the dog then you may very well be allergic. An allergist can confirm this for you or if you notice your symptoms worsen around the dog, then you might have your answer. If so, zyrtec/claritin may improve your symptoms.