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 even temporarily increase your Pokémon's stats. However, this mechanic is missing from the latest pair of Pokémon games, Brilliant Diamond and Brilliant Pearl. Since Z-Crystals are not in the game, there is no way to perform Z-Moves in the newly released remakes.
The exclusion of the Z-Moves by the remakes, along with other mechanics such as Dynamaxing and Mega Evolutions, can probably be attributed to these mechanics originating from the Pokémon games released 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 combat system works exactly like in the original games. Interestingly, the game includes minor innovations from Generation IV games, such as Fairy Typing for Pokémon.
Since Z-Moves were only present in Generation VII games and didn't appear in Pokémon Let's Go games or Pokémon Sword and Shield, if the mechanic returns it's pending. For now, the only Pokémon games where you can use Z-Moves are Sun, Moon, Ultra Sun, and Ultra Moon.