After Quicken resolves, you can Suspend a sorcery in your hand any time you can cast an instant. As soon as you actually cast a sorcery, you lose this capability.

You don't choose a sorcery card when Quicken resolves. Rather, this sets up a rule that is true for you until the turn ends or until you cast a sorcery card, even if you cast that sorcery at a time you normally could.

If you cast multiple Quickens on the same turn, they'll all apply to the very next sorcery spell you cast.

Quicken : règles, fonctionnement et interactions (MTG)

Quicken is one of those cards players keep asking about. Here is, in plain words, what its official text says, the rulings published by Wizards of the Coast, and the combos players actually run.

Official card text

The next sorcery spell you cast this turn can be cast as though it had flash. (It can be cast any time you could cast an instant.) Draw a card.

How it works : Quicken

Original English text — Wizards of the Coast / Scryfall

#1September 25, 2006· Wizards of the Coast
After Quicken resolves, you can Suspend a sorcery in your hand any time you can cast an instant. As soon as you actually cast a sorcery, you lose this capability.
No other card is named in this ruling.
#2February 1, 2006· Wizards of the Coast
You don't choose a sorcery card when Quicken resolves. Rather, this sets up a rule that is true for you until the turn ends or until you cast a sorcery card, even if you cast that sorcery at a time you normally could.
No other card is named in this ruling.
#3February 1, 2006· Wizards of the Coast
If you cast multiple Quickens on the same turn, they'll all apply to the very next sorcery spell you cast.
No other card is named in this ruling.