We all know that vibration is the cause of sound generation. But here my question is, Internally what happens in vibrating object which creates sound? I guess, generation of sound will be some where related to atomic level since any form of object (Solid, Liquid, Gas) can create sound.
The most common approach to model the sound radiation from a vibrating body is generaly the same as in all wave cases: continuity is the key. Let's say that a sphere is vibrating (changing it's volume periodically), then the acoustic velocity of the air particles just on the boundary with the body must be the same as the velocity of the sphere surface. It gives us boundary condition for solving the wave equations.
If your question is ment to be more focused on the vibrating body and aiming for atomic level, then phonon should be your keyword. But in that case your question is way too broad and it should be rephrased.
May I ask what you think of as sound?
If sound is the vibration of air- or in general any material agent- then sound is the sensation you get from the changes in the pressure of the air, it's what reaches your ear and then produces some signals interpreted in your brain. Sound is the vibration, not something produced by the vibration.
This vibration which is called by us sound is a mechanical wave define as a vibration in pressure or more general in particle displacement.
Hope this helps with your question.
Take your hand and move it. By moving your air, you moving the air. This is what a vibrating object does - it moves the air. Sound is just the movement of air (or a liquid or solid).