QCan a follower or amulet with a "Q" ability also be played when an enemy follower attacks or during my opponent's end phase?

ANo. The "Q" icon is tied to a specific card ability, not the card itself, so the follower or amulet cannot be played as if it had Quick.
An ability with a "Q" icon can be played with the same timing as Quick when in the appropriate zone.

QWhat does the "Q" icon mean?

AThe "Q" icon indicates that the ability can be played with Quick timing.
Abilities with this icon, like cards with Quick, can be played like usual during your main phase, as well as when an enemy follower attacks or during your opponent's end phase.