Position a rack in the center of the oven and preheat the oven to 350°F. Generously butter the bottom and sides of a 13 x 9-inch baking dish.
Place the oats, flour, flaxseeds, baking soda, baking powder, cinnamon, and salt in a medium-size mixing bowl and stir to combine.
Add the oat mixture to the buttermilk mixture, and stir to combine. Add the carrots and apples, and stir just until blended. Transfer the batter to the prepared baking dish.
Place the walnuts, coconut, brown sugar, and cinnamon in a small bowl. Stir to blend, and sprinkle the topping mixture evenly over the batter.
Bake the squares until the batter has set and a toothpick inserted in the center of the cake comes out clean, about 45 minutes. Let cool on a wire rack for 1 hour. Then cut into 12 pieces. (The squares can be stored in an airtight container at room temperature for up to 4 days.)