No, but a cyst could. Ovaries don't pop from blunt trauma, because they are small (size and shape of a large almond), are solid, and are deep in the pelvis. However, an ovarian cyst (a liquid-filled lump on the surface of an ovary) that keeps growing can get to several inches in diameter. Such a cyst could "pop" from trauma, which would likely cause great abdominal pain with internal bleeding (and a trip to the E.R.).
Unlikely. Ovaries don't really "pop" in the sense like water balloons and they are deep within the body cavity under protection. It is more likely direct trauma will hurt testes because they are on the outside of the body cavity rather than ovaries (which are small and protected by layers of tissue).