Heat olive oil in a large skillet and add in chopped beef. Cover, bring it to the lowest heat and let it cook until first release juice and then absorb most of it. Toss in sliced onion, green and red peppers. Add the spices. Cover and let it cook over lowest heat until tender, about 30 minutes. If it gets too dry, just pour a little hot water and continue cooking until tender. Let it cool.
Place some shredded cheese on one tortilla. Then spread some beef filling over it. Toss some chopped spinach leaves and more cheese over it and put another tortilla over it. Gently press over it so that there are no air pockets.
Brush a non-stick skillet with a little olive oil and heat it. Cook the prepared quesadilla in it until golden brown on both sides, about 5 minutes for each side. Repeat for the other tortillas.