Since I am on a blood thinner, would taking Zyrtec cause my bloody nose that I am experiencing?

It very well could. The Zyrtec is probably drying the nasal passages out causing them to crack and bleed a bit. Using a little saline spray to moisturize might help along with putting just a teeny dab of petroleum jelly inside each nostril. Always remember though that nosebleeds can be a sign that your INRs are too high so please notify your Dr to see if he thinks you should have your blood work checked again.