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No and (maybe) Yes.

No. Current is the movement of electric charge from one point to another under the influence of an electric potential difference. You can have charge without current.

(Maybe) Yes. You can also argue that charge is constantly moving at the atomic level as the electrons move between their various quantum states, but you can also (maybe) argue that this is not in response to an electric potential difference. You can also argue that, once you back out even a little bit and broaden your point of view, the net effect of that movement is zero.

The normal understanding of current is a daisy-chaining of electrons jumping from atom to atom at the valence shell level under the influence of a much more global electric potential.

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