I don't use Bullet, but looking at the documentation, I think this should work:
btQuaternion diffQuat = (currentQuat.inverse() * FinalQuat).nearest(btQuaternion(1.0, 0.0, 0.0, 0.0));
if (diffQuat.getAngle() > max_angle))
diffQuat.setRotation(diffQuat.getAxis(),max_angle);
btQuaternion newQuat = currentQuat * diffQuat;


That code uses a couple of trigonometric functions more than strictly necessary, but given the interface of btQuaternion, it seems to be the easiest way to do it.

Let me know if that works.

It's not working, basically I want to animate the rotation of a tank cannon, the tank cannon should rotate with certain speed, so the method can look like that:

// Every call will rotate the cannon according to 'rotationSpeed'
bool lookAt(float x, float y, float z, float rotationSpeed, float elapsedTime);



Here is the idea:
1. The tank see the enemy
2. The tank rotate it's cannon towards the enemy lookAt(enemy.x, enemy.y, enemy.z, 0.1f, elapsedTime);
3. Keep calling lookAt() each frame until lookAt() return true, when lookAt() return true, fire a missile towards the enemy.