Preheat the oven to 350 degrees F (175 degrees C). Grease a 5x7-inch glass dish.
Mix ham, tomatoes, salsa, 1/4 cup milk, Cheddar cheese, jalapeno Jack cheese, cilantro, and green onions.
Place half of the tortillas into the prepared baking dish and top with half of the ham mixture. Repeat layers. In another bowl, mix eggs, 3/4 cup milk, salt, and pepper. Pour over mixture in the baking dish.
Bake in the preheated oven until top starts to brown, 60 to 75 minutes. Remove from the oven and let sit for 10 minutes before cutting. Sprinkle with cotija cheese.