Preheat the oven to 375°F (190°C). If using a store-bought pie crust, place it in a 9-inch pie dish and prick the bottom with a fork to prevent bubbling.
In a skillet over medium heat, sauté the chopped onion in a tablespoon of olive oil until translucent, about 5 minutes. Add the minced garlic and cook for another minute until fragrant.
Add the grated zucchini to the skillet and cook for 3-4 minutes, stirring occasionally, until slightly softened. Remove from heat and let the mixture cool.
In a mixing bowl, whisk together the eggs, heavy cream (or milk), salt, black pepper, and dried thyme. Stir in the shredded cheese and the cooled zucchini mixture.
Pour the filling into the prepared pie crust, spreading it evenly. If desired, sprinkle fresh parsley on top for added flavor and color.
Bake in the preheated oven for 35-40 minutes, or until the quiche is set and lightly golden on top. A knife inserted in the center should come out clean.
Allow the quiche to cool for 10 minutes before slicing. Serve warm or at room temperature.