Quests in Hollow Knight are considered as side-quests or tasks that are given by NPCs. Quests provide the players additional information in order to learn more about the story of Hallownest - these quests will provide the player rewards upon successfully completing the quest.
Hollow Knight Quests
"Ull Waaaaii! Che's sacred, precious, one of its kind flower. Ruined. Che's love, denied her final gift."
A Nailsmith's Obsession
"With a Pure Nail forged, my work in this lifetime comes to an end. My only remaining desire is to see and feel the nail strike true!"
"Ah! The way you hold your nail! So you've learnt a Nail Art..."
Dream No More
Secret of the Stagways
"Continue to open the stations and you'll have my deepest gratitude."
"I... you came to rescue me, Bretta? The girl that everyone just... ignores?"
"Losing one's Geo is a terrible thing and it's so much more likely if you carry it about on yourself."
The Grimm Troupe
"When flames burn bright inside one’s mind, kin heed the call ‘cross lands and time.”
This tells me nothing about the quests!