Yes, grass of any kind can cause allergies , as well as ragweed , pollen and dust. Generally you will see itching, scratching the ears, rubbing the muzzle along the ground or furniture an licking of the paws. These are indications or airborne allergies.
Food allergies would show scratching, a rash on the belly area or inside the leg area. The majority of food that cause a reaction are chicken, beef, lamb and grains. Dogs usually show a reaction to food they have been eating for a while, not ones that are recently introduced.
Airborne allergies: http://www.petplace.com/dogs/seasonal-allergies-and-your-pet/page1.aspx
Food allergies http://www.petplace.com/dogs/food-allergy-in-dogs/page1.aspx