There is not the slightest evidence that Q's powers are somewhat related to the Spore Drive. Some random guy's twits don't really count, I fear.
Q, both as a single individual as portrayed by John De Lancie, and as a whole species, are featured prominently on The Next Generation, and also make some appearance on Deep Space Nine and Voyager.
In all these appearances, they are always shown to posses godlike powers, like yourself are acknowledging in your questions. Going instantly in any given point of the galaxy is among these powers, but it is certainly not the only one: they can change their appearance at will, they can time-travel, they can transport other beings (even whole starships) trough time and space without effort, and trick you in many other ways if you let them:
Briefly, we can summarize all of this by saying that physics laws are not constraining them at all, and that with their powers they can effectively do whatever they please. The origin of their powers is never described in detail but it is implied that they possess them inherently, not as the result of an incredibly advanced level of technology; and, more to the point of your question, it is never stated, suggested or even subtly implied that they use a spore network to travel.
We know that Spore Drive is a FTL method that just allows incredibly fast speeds, it does not completely suspend all physics laws for its users. So, it can't be accounted to be the source of all other Q powers, and it is highly improbable that they, having godlike powers in all other respects, are constrained to use a physical mean just to travel.