Can you exhaust a unit without moving?

No — you can't just exhaust a unit for no reason. Exhausting is a Limited Action, so you only do it when a rule or effect tells you to: the Standard Move (exhaust as the cost), an activated ability with an exhaust cost, or a card that says "exhaust a unit" (Rule 415.3).

So exhaust is almost always tied to some action — most commonly the Standard Move, but you can also exhaust a unit to pay an activated ability's cost without it ever leaving its location. The thing you can't do is rotate a ready unit as a free no-op (FAQ #7132).