NO. Cold weather cannot give you allergies.
However when you're inside more often due to the cold weather you can get our allergies to flare up because there could be allergens in the house that you're breathing in.
So you may be suffering from allergies during the winter months but it's not because of the cold weather though.
Dust and other allergens can be in the air and in your furnace air filter that get thrown around in the house when the heat is on.
During the winter allergens in the house such as dust mites, mold, and pet dander.
So you may suffer from allergies during the winter time but you may also suffer from allergies in the spring and summer as well.