The terms "activation" and "acting" can be a little misleading, I suppose, but luckily they are not actually used in the official rules for Daze. Per the codex entry of Daze, "All Daze markers are removed at the end of the creature’s Action Phase."
For a full run down of what Action Phase means, you will have to read through the entire rule book, as it is mentioned in several places. A quick summary of it though is that it begins with Activating the creature (i.e. flipping it's Action Marker over) and only ends after it has completed doing everything it wishes to do and you are ready to pass turn to your opponent. It is at this point, the end of the creature's Action Phase, that Daze is removed.
The Quick Cast Marker is not an Action, and is never used during an Action Phase. It is used Before or After a friendly creature's Action Phase. Thus, using the Quick Cast Marker will never remove Daze.