The card Quicken has the effect
The next sorcery you cast this turn can be cast as though it had flash.
Does that mean I can cast my next sorcery during my opponent's turn?
Yes, that's what it means.
In addition to the quoted rules text, Quicken has useful reminder text:
It can be cast any time you could cast an instant.
This is a shorthand that really means you need only priority to cast it.
304.5. If text states that a player may do something “any time he or she could cast an instant,” it means only that the player must have priority. The player doesn’t need to have an instant he or she could actually cast. Effects that would prevent that player from casting a spell or casting an instant don’t affect the player’s capability to perform that action (unless the action is actually casting a spell or casting an instant).
Now, what Quicken actually does is letting you ignore the normal timing rules for casting a Sorcery, and instead allow you to cast it any time you have priority, as long as it's within the same turn.
Whether you use it to cast a Sorcery in your own turn when you normally couldn't cast a Sorcery (for example in your Upkeep or End Step), or during an opponents turn is completely up to you!
It depends on whose turn it is when you cast Quicken.
Look carefully at the card's text (emphasis mine):
The next sorcery card you cast this turn can be cast as though it had flash.
Quicken sets up a rule that is true for you until the turn ends or until you cast a sorcery card. If you cast Quicken on your turn, then the effect will wear off before your opponent's turn. But if you cast Quicken on your opponent's turn, then you can cast a sorcery during that turn.