There are a number of things which are problematic with identifying infinity with zero, with entertaining the idea that they are equal.
The first you should ask what you really mean by equality. How serious are you about the idea that "zero" and "infinity" refer to the same concept? How many Bengal tigers are there, for instance, in your immediate vicinity — none, or infinitely many? (If there happen to be one or two, would you expect there to be infinitely many of them if those two went away?) I would suspect that there being zero Bengal tigers nearby to you would appear very differently to you than there being infinitely many. This gets to the heart of what mathematics is actually for: it's for describing features of the world that you see around you — and for being able to express different ways that the world could be, which you could distinguish from one another if you wished to. If it were somehow "really the case" that zero and infinity were equal — that is, if this somehow were a deep meaningful feature of the world — why are we not being constantly assaulted by (or at least crushed to death under the weight of) infinitely many Bengal tigers?
A better response might be to say that "zero" is not the same as "infinity" on the level of tigers or birds, but only for other physical phenomena, such as matter in the universe. (This would already indicate that the two concepts of zero and infinity are meaningfully different, and that what we're talking about is not mathematics, but physics proper.) Perhaps it is only a meaningful way of describing things on the quantum mechanical level. But here it is no better: why are there not infinitely massive balls of neutrons and protons and electrons popping out of vaccuum — not just a handful with some probability, but infinitely many, all the time, because "nothing is the same as everything"? Sure, it would crush the universe to a tiny speck under the instantaneous emergence of black holes all over the place; but this just gives us a way to see that it isn't happening, not of explaining why it doesn't if somehow "zero" equals "infinity". The problem is that even if you restrict yourself to "the quantum mechanical scale", saying that "zero equals infinity" doesn't allow you to describe features of the physical world with enough precision to explain why at any moment we aren't consumed by black holes.
Note that the statement "on a quantum mechanical scale" is itself a vague statement. Most physicists believe that matter behaves according to quantum mechanics at all scales, it's just that for objects which are large and rigid enough, we can use less complicated models of physics such as Newtonian mechanics to describe what's going on. So even saying "on a quantum mechanical scale" is insufficient to save us from an infinite avalanche of bengal tigers. If we want to put teeth into a statement such as this, we need something subtler than equality; we need actual numbers and differences, to describe differences in size and in probability.
This is another problem which is touched on by your question. Quantum mechanics does indeed depend on "randomness"; but randomness is not the same as "anything can happen". For instance, a dice roll is random: but would you expect to roll a seven on a single die, because it is random? More precisely, not everything that can happen will actually happen: if you rolled the die a thousand times and only rolled sixes and twos, wouldn't you come to suspect something was wrong? But certainly it's possible. The problem with the die which rolls only sixes and twos is that it violates your expectations, which is a way of observing that there are limits and averages which you can expect from the die. Similarly, although quantum mechanics is random — and also to our macroscopically-honed expectations, strange — this does not mean that it's a free-for-all of strangeness at all times. The very fact that we have a theory of quantum mechanics that works at all, indicates that it has regularity and predictability about it; it only has less predictability than a deterministic theory of physics in which we can finely control the initial conditions of the system.
So what could it mean for "zero" to be the same as "infinity"? Well: all numbers — including simple ones, such as 1, 2, 3 — are just ideas, and they can mean different things in different contexts. When you roll a die, the number 6 doesn't mean anything, although you might give that number signifcance by doing something specific. In the game of craps in which you roll a pair of dice, 7 and 11 are good rolls and 2, 3, 12 are bad ones; but that does not mean that somehow 7 = 11 or that 2 = 3 = 12 in any deeper sense. These are just human games, of course; but numbers are human ideas with which we try to grasp the world with acuity. The roles of any numbers in a physical theory do not arise from the numbers themselves, but from their interpretation as referring to magnitudes of physical qualities which interact with one another.
So the only way to assess whether or not zero is "equal to" infinity in some physical theory is to see whether the two concepts are effectively the same in that theory, at least for the main quantities of interest. For the physical theories that I know of, the answer is a resounding no. This does not mean that there could not be another, very useful theory in which some quantity such as mass or time somehow might be meaningfully interpreted in such a way that zero equals infinity, but that will be a question of that model of physics, and not of reality itself. In the end, zero and infinity are the names of ideas, of metaphors that we use to grasp the world — they are the map and not the territory.
(If you want to ask about whether the lack of any existence is identical to everything existing at once, without reference to mathematics, that would be a separate question, but one for which I doubt that there is an interesting answer; I would say it is either "no" or "there's no way to find out", depending on what limitations you put on the multiply existing worlds.)