What to do for if I do not cough, but it sounds like I have a slight wheezing sound through my nose. Does this sound like it could be asthma?

Nasal obstruction. If the wheeze occurs at the nasal area, then it is likely nasal obstruction. Do you still have this wheeze when you breathe through your mouth? Do you have any shortness of breath because of this? We need more information to venture a guess.
Maybe. Wheezing is a common symptom of asthma but not in the nose. This should be checked by a physician and if necessary referred to a specialist (ears, nose and throat). Good luck.