No, because you are traveling at the same speed as the Earth. It's the same as if you are inside of a car, and you have something in your hand and you toss it straight up into the air, it will fall right back down into your hand, not hit you in the face. Or, if you are standing on a train or bus and jump while it is moving, you will land in the same place because you are moving with the vehicle. The first experiments with this sort of thing were done by dropping weights from the tops of masts on sailing ships. The ship is moving forward, sometimes very quickly, yet the weight always falls at the base of the mast when dropped.

Now, if you were able jump high enough into the air, you would start to noticeably lose some speed by the time you got back down to Earth so you wouldn't land in the same spot. But at the heights we are capable of jumping, there's no noticeable difference. You'd really need to go quite high before this could be noticed. Even dropping objects off of a tall building doesn't cause a noticeable effect and you're much more likely to have alterations in path of the falling object due to wind than the movement of the Earth.