If an object was dropped into water from two different heights, assuming that it doesn't hit the bottom, would the terminal velocity be different?". Suppose that container has a lot of depth that it is not the issue to reach terminal velocity without hitting the bottom of the surface of water container?
Since drag force varies with velocity, the terminal velocity of an object merely depends on the properties of the liquid and the shape of the object (should it matter). check https://en.wikipedia.org/wiki/Terminal_velocity (section Physics). The answer is no, it doesn't depend on initial velocity.