Would osmosis occur between two solutions of equal moles of water but unequal moles of a nonpolar molecule?
Your example with the no-polar substance wouldn't qualify in a discussion of osmosis. There needs to be a solution on at least one side of the membrane. Since a typical non-polar substance would be insoluble in water, osmosis would not apply.

However, in your diagram, water would move from left to right due to a concept called pressure potential. You will eventually learn about this in AP Bio (I taught it for 24 years.) Pressure potential and osmotic potential work together, in a concept called water potential, to calculate which way water will move across membranes. You will probably perform a lab utilizing this concept this year. Pressure potential, in your example, is just a fancy term stating that because both the atmosphere and the non-polar substance on top of the left side are greater than just the atmosphere pushing down on the right, water will move to the right.

One of the things you should be careful about is that it is not the number of moles that are important, it is the concentration of substances. Get rid of your non-polar substance. Move the membrane over to the left so there is much more water on the right. Imagine the water levels are equal on both sides so we don't have any pressure potential to deal with. In this scenario, there would be no net movement of water even though there's far more total water on the right. This is because the concentration of water on both sides is the same, 100%