In the heavy or long projects when I go to Piano roll and right click on any midi note to change the velocity manually, for some reason that midi note slightly moves to the right and I have to quantize it again. I tried it on a new project, it is fine. I realized it just happens on the heavy or long projects. 
Any help would be appreciated.
 

sounds like a problem with snap maybe ?
As a test, try turning snap off and doing it.

If this is X1 or later, you must be using something other than the Smart Tool or right-click would delete the note instead of bringing up properties. Which tool? Right offhand, I'm not aware of any thing that would cause that to happen, especially if the note is already quantized to the snap resolution.
 
Try editing velocity with the Smart Tool by the Velocity hotspot on the upper edge of the note instead.