Calcite is one of the more stable forms of calcium carbonate, and quartz is an extremely common mineral made of silicon dioxide. Out of the both of these, I'd rather buy jewelry made from quartz, and it mostly comes down to durability.
First, let's take a look at hardness. Quartz has a mineral hardness of 7, known in official terms as "very hard". Meanwhile, calcite has a hardness of 3, which is pretty soft in relation. So if you have jewelry made from calcite, you'd want to be careful to bump it against pretty much anything, because it'll chip and break off pretty easily.
Additionally, calcite dissolves in acid quite easily, putting a little acid on a mineral is one way to taste whether or not it's calcium carbonate. Even the acidity of carbon dioxide in the air or mildly acidic rain could lead to some dissolving, not something you'd want for an expensive piece of jewelry, right?
Finally, we can look at cleavage. Different minerals will cleave in different ways and directions, often giving chunks of them a characteristic shape. Calcite cleaves in three dimensions, meaning that, structurally, it is weak in all three dimensions. Meanwhile, quartz has no cleavage planes in any direction due to it's molecular structure, so it's less likely to cleave in any direction, it'd just fracture.
Hope this helps!