Probably no, which surprises me a little.
The relevant rule is that
"You can reveal an enchantment immediately after
any action or event in the game:"
We know what actions are, but what are events? The only definition of events I can find occurs at the bottom of the sidebar:
You cannot interrupt an event to reveal an enchantment. Example: You cannot reveal an enchantment on a creature in the middle of its Move Action, or in the middle of rolling dice during an attack. You would have to wait until that “event” (step or action) has finished.
Which is weird, because that rule can now be re-written as
"you can reveal an enchantment immediately after any action or step or action." Something odd's happening here.
Anyway:
We know that you can reveal an enchantment after a creature is activated, before it chooses its actions for the turn (because this sidebar says so), but the reasons for that aren't clear, because flipping an action marker is neither an action (like moving or making a full attack) nor a step (like step 3 in an attack rolls dice). So why is flipping an action marker an event?
The rules aren't clear on this, other than "because it says you can."
However, nowhere in the rules can I find any statement that says you can reveal an enchantment after flipping a quickcast marker but before the quick spell, and the rules are at pains to point out the many other opportunities to reveal an enchantment.
So, what alternate interpretations might let you reveal the enchantment before the spell?
1. If flipping an creature's action marker is an event, even though it's not an action or a step, maybe flipping a quickcast marker is too.
2. The rules don't list the steps involved in taking the quickcast action. It's possible that there are steps in taking a quickcast (as there are steps for attacks, spells, and movement). If this were the case (and there's no written reason to think this would be true except consistency) you could be revealing the enchantment after the 1st step, which would be an event.
3. Alternately to #2, the rules don't actually say when the Quickcast action ends. I've always assumed that the quick spell occurs during the quickcast action, but the rules just say that taking a quickcast action lets you cast a quick action spell. It's possible, but not consistent, that the rules literally mean that the mage takes a quickcast action which ends, which then lets the mage cast a quick spell. If this were the case, you could reveal an enchantment after the quickcast action, but before the quick spell. I think this interpretation is grasping at straws, but the rules as written are sufficiently silent about the nature of the quickcast action that it's not precluded by anything but the principle of parsimony.