It's very hard to tell from the description, but I think my 10 Roadster makes a simular noise. It was also present on my other 2 350Z roadsters. The top clamped tight on them, so I always figured it was the locator dowels on the doors. It was especially present anytime after it had rained or been very humid. It was like a thin coat of rust would build up on the dowels and cause a squeak or rattle until the rust got knocked off. Then it would clear up for a while. Often a long while depending on the weather. The 370 makes the same sound. It only stands to reason that there is a lot of stress on those dowels as there's no roof to stiffen the car body and even though it has stiffeners on the chassis, it still uses the doors to assist. Also where the 350 had clamps that pulled it down tight, the 370 has dowels up there too that it uses to clamp the roof down too. It's just a suggestion, but maybe if you put a thin coat (very thin) coat of lubricant on all of them, it might quiet down just a little. The worst case senario is that it changes nothing and you can rule that out.
I hope you get it figured out. A noise like that can drive a person nuts. For me, that's a very short drive.
But the truth is, I enjoy this car too much to let it's little gremlins spoil that for me. I love this car.