I started eating greek yogurt & I noticed thicker mucus in my throat. Is this supposed to happen?

Milk allergy. Could be that you have allergies to milk products. Try soy yogurt and see if that helps.
Not really. Studies have not confirmed a relation between dairy and mucus. The problem may be consistency (strained, denser product than regular yogurt), reflux of stomach contents into esophagus which could increase mucus production, fat content (greek yogurt comes 0%-4% fat), protein content (twice regular yogurt or milk) unlikely to be allergic reaction. Drink more fluids; see if it helps.