5th Edition doesn't have a [mechanical] distinction between Arcane and Divine spellcasting
Each spellcasting class has different rules regarding what is allowed to be used as their spellcasting focus, and Warlocks (and Sorcerers and Wizards) are allowed to use an "Arcane Focus" as their spellcasting, suggesting that they constitute Arcane spellcasters. This is also confirmed by the Lore provided in the Weave of Magic blurb found in the player's handbook:
All magic depends on the Weave, though different kinds of magic access it in a variety of ways. The spells of wizards, warlocks, sorcerers, and bards are commonly called arcane magic. These spells rely on an understanding—learned or intuitive—of the workings of the Weave. The caster plucks directly at the strands of the Weave to create the desired effect. Eldritch knights and arcane tricksters also use arcane magic. The spells of clerics, druids, paladins, and rangers are called divine magic. These spellcasters' access to the Weave is mediated by divine power—gods, the divine forces of nature, or the sacred weight of a paladin's oath.
—The Weave of Magic, PHB, pg. 205
But mechanically, this doesn't matter. There's no such thing as "Arcane Spells" or "Divine Spells" in 5th Edition D&D, with any kind of mechanical impact on how the spell is cast or how it functions. Spells are just spells, and individual classes have individual rules regarding how they cast spells and which spells they're allowed to learn and how they "prepare" spells (if they prepare spells at all).
The Dark Shard Amulet calls for an "Arcana" check, not an "Arcane" check
Arcana is a skill that a character may or may not have proficiency in, that usually (though not always; see PHB pg. 175, Variant: Skills with Different Abilities) is based on a character's Intelligence score. This skill is accessible to any class in the game. So if you attempt to use the feature of the Dark Shard Amulet, you're called upon to perform an Intelligence check that allows you to use your Arcana skill proficiency, if you have it, in order to perform the Cantrip you don't already know. The happenstance of the source of your magic being Arcane (or not) is immaterial.