In gioielleria ho visto una piccola gemma verde che brillava sotto la luce.

In Italian, many shops and public places pair in with the bare noun to express “in/at a …” generically. Examples include in banca, in chiesa, in università.
In gioielleria follows this pattern, meaning “at a jewelry store” in general. You could say alla gioielleria (“to/at the jewelry store”) when you refer to a specific one, but in gioielleria is more idiomatic for the general location.

The passato prossimo (ho visto) is the standard tense in spoken and most written Italian to indicate a completed action in the past.
• Vidi (passato remoto) is reserved for formal/written narratives or certain regions.
• Avevo visto (trapassato prossimo) describes an action completed before another past event, which isn’t needed here since there’s no earlier reference point.

Che is a relative pronoun linking una piccola gemma verde (the antecedent) to brillava, introducing the relative clause. You cannot simply omit it—you need a connector. For a more formal style you could replace it with la quale:
una piccola gemma verde, la quale brillava sotto la luce

When referring to a specific condition of “the light” shining on something, Italian uses the definite article. Sotto la luce literally means “under the light.”
Dropping the article (saying sotto luce) can occur in technical or stylistic phrases (e.g. osservare sotto luce polarizzata), but in everyday speech sotto la luce is the natural choice.