Heat 1 1/2 teaspoon olive oil in a nonstick oven-safe skillet over medium heat; cook and stir green onions and garlic in the hot oil until the garlic is fragrant but not browned, about 1 minute. Stir in spinach, cover skillet, and cook until spinach is wilted, about 5 minutes. Transfer spinach to a large bowl.
Heat 1 1/2 teaspoon olive oil in same skillet and turn heat to medium-high. Cook and stir mushrooms until lightly browned, about 5 minutes; remove from heat.
Squeeze any moisture from spinach mixture and stir into mushrooms. Beat eggs in a bowl until thoroughly combined. Stir eggs into the spinach-mushroom mixture; thoroughly stir in feta and jalapeno yogurt cheeses.
Spoon the mixture back into the skillet; use a spoon to level surface. Wipe any filling from edge of skillet above the filling with a paper towel.
Bake in the preheated oven until the quiche is golden brown, about 45 minutes. Let cool slightly before cutting into slices.
