Has there been any effort to understand Hubble's limit as a quantum phenomenon?
It seems like they are suggesting the same thing.
Hubble's constant is derived from General Relativity, and specifically from the Friedmann equations. The parameters needed are all macroscopic e.g. the average density of the universe so it isn't obvious that quantum mechanics plays a role, except of course that General Relativity is probably the large scale limit of some theory of quantum gravity.
Were you thinking of dark energy? There have been various suggestions about linking this to the zero point energy of quantum fields, or more recently relating it to local minima in the String Landscape. However this is all pretty speculative.