Considering tht the feat specifically states "You gain an attack of opportunity whenever an enemy attempts to grapple you".... the answer is still no... Close-Quarters Fighting does nothing that normal grappling rules don't already do besides counter Impproved Grappling... They over-elaborated the description for it, but even in their over-elaboration they still did not open it up for anything else.
Entering a grapple triggers AoO, Improved Grapple denies AoO, Close-Quarters Combat denies Improved Grapple & adds damage to opposed grapple check when against Improved Grapple. Leaving a grapple triggers an AoO to those NOT involved in the grapple if one of the grappling party moves to exit the grapple. Maintaining a grapple triggers an AoO to those NOT involved in the grapple when the grappler moves into the target's square. The target of a grapple only gets an AoO when the grapple is initiated, period.