Z-Moves were introduced in Pokémon Sun and Moon, where they functioned as super-moves that could be performed once per battle. By equipping certain Pokémon with certain elemental crystals, you could unleash an often-devastating attack that would also boost your Pokémon’s stats temporarily. However, this mechanic is not present in the latest pair of Pokémon games, Brilliant Diamond and Shining Pearl. As Z-Crystals are not in the game, there’s no way to perform Z-Moves in the newly released remakes.
The remakes’ exclusion of Z-Moves, alongside other mechanics such as Dynamaxing and Mega Evolutions, can likely be attributed to these mechanics originating in Pokémon games that came out after Pokémon Diamond, Pearl, and Platinum — the trio of games on which the remakes are based. Mechanically, Brilliant Diamond and Shining Pearl are nearly identical to their inspirations; the battle system works exactly as it did in the original games. Interestingly enough, the game does include smaller innovations originating from post-Generation IV games, such as the Fairy typing for Pokémon.
Given that Z-Moves were only featured in the Generation VII games and also appeared in neither the Pokémon Let’s Go games nor Pokémon Sword and Shield, whether the mechanic will return is up in the air. For now, the only Pokémon games you can use Z-Moves in are Sun, Moon, Ultra Sun, and Ultra Moon.