Doesn't quantum mechanics tell us that something could come out of nothing? Then the universe wouldn't require God.

Well, when we're talking about quantum mechanics, we have to be careful about what we mean by nothing. Do we mean nothing as in "not anything, like "I ate nothing for breakfast" or do we mean "nothing" the way air has "nothing in it" (even though there is technically something there like oxygen and nitrogen)? Because when scientists talk about the universe coming from nothing, they really mean nothing in the second sense, the nothing of empty space. And according to our current physics, empty space is technically not nothing, it has energy and properties. Scientists like Sean Carroll and Lawrence Krauss talk about this type of "nothing" as something unstable, having energy, having a certain behavior, and even weighing something. In other words, that nothing is something! So this theory doesn't actually allow for the universe to pop out of truly nothing.

But there's another way that that the "quantum nothing" hypothesis challenges the argument. Since the quantum nothing is obviously a part of the universe, albeit before the universe we see now, it subtly rejects premise #2, that the universe began to exist. Because sure, even if the universe we see was created by quantum effects, maybe the empty quantum vacuum itself and the laws of physics that describe it were not created. They were just there, and they've always been there, and so the universe never began.

But actually, this doesn't work. Scientists reject this theory now because it would make no sense for a quantum vacuum to exist forever and then at some particular point, to spontaneously create a universe. If it existed forever, why suddenly create a universe 13 billion years ago? So while it might be possible that the universe was created by a quantum event in empty space, that empty space isn't really empty, and even that "nothing" of sorts would require a beginning as well.