Can quadcopters still fly with propellers at an angle?
Yes, you can. But it very depend with the angle (as in your drawing). If it is "folded) to make V-shape (that will make divergent thrust), then will reduce the lift compared to if the angle is opposite direction (inverted V-shape), which will make the thrust become convergent, it will produce more thrust. Both will work, but one is better than another. See this Akagi Helicopter. But the angle should not be high enough.

There is research that researching eagle and bird shape. There are two kind of bird or eagle's shapes. The 1st is the wing feathers trailing edge is divergent, which is make it look like V seen from the head. The 2nd is, the wing feathers trailing edge is convergent (closing to the tail, so it look like V seen from the tail, not from the head). From the research, the 2nd is better than the 1st in maneuvering and speed. But in slow move (like surveillance), the 1st is better as the flaps are produce less speed and power for the same amount of flapping (power). Another thing, the power of the flapping of the 1st is divergent, leaving the bird/eagle body, while the power of the flapping of the 2nd eagle is concentrated beneath the body (convergent) that make amount of lifting power.  Unfortunately I can not find the video in Youtube. Maybe later.