In a large bowl, measure the flour, baking powder, cinnamon, baking soda and salt. Stir to combine all the ingredients well. Make a well in center of the mixture.
In a medium bowl, combine the eggs, sugar, oil and ginger root. Add the quinoa, grated carrot and apple and stir. Add to this to the flour mixture. Stir until just moistened. Spread the batter evenly in 9 inch (22 cm) greased springform pan.
Bake in preheated 325ºF (160ºC) oven for 55 minutes or until cooked through. Let cool for 5 minutes. Invert cake onto wire rack to cool completely.
To Make the Icing:
In small bowl, beat the cream cheese, milk, butter and vanilla until smooth. Add the icing sugar. Beat until smooth. Add more milk if necessary.
Spread evenly over top and side of the completely cooled cake.