After pondering it and doing a little research, I'm inclined to say: "don't be silly, of course not!"
Therefore, it's likely that plants don't get gallstones either.
However, plants can get galls, each of which consist of a swollen, lumpy growth of plant tissue, similar to a benign tumor that a human would get.
For example, if some insect lays its eggs on the leaf of a plant, the plant may form galls to act as a way to protect itself from the irritation. And, also, the insect gall can serve as a little microcosm and a source of food for the insect. The insect larvae can remain inside the gall until they are fully grown and emerge.
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