JJR, I have an 07 Qx56 that had about 170k on it, and had all the same symptoms yours has. I thought it must be a dying battery so swapped and still had intermittent no-starts. Finally took it to a good indie shop and they traced it to a bad starter. Replaced that and it's all good since! Cost around $700 because it's located under the intake manifold, so it's a bit of a plain to get at. Good luck, let us know what it was when you get it fixed!