The word "quantum" refers to the absolute smallest single thing that can exist. A photon, for example, is one quantum of light, since you can't get any less light than a single photon. Along with light, you can also have a quantum of sound, called a quantum phonon, and by definition, it's the quietest sound in the universe.
At its most basic, sound is energy that's transmitted through a material (including air, water, or just about anything else) in the form of a mechanical vibration of matter. For example, when you inform your co-workers about how that taco you just ate for lunch was a little bit funky and that they should expect the worst, your vocal cords are creating subtle pressure waves that cause air molecules to vibrate back and forth in a pattern that our eardrums can detect and translate into information. This, incidentally, is why in space nobody can hear you scream: there's nothing in a vacuum to vibrate.
Sounds we can hear generally consist of lots of molecules vibrating in big waves, but the principle applies at much smaller scales as well. You can transmit a sound through one single atom if you can get it to vibrate back and forth in the pattern you want, and there's no reason why you can't go even smaller than that, since the amount of back and forth motion that the atom can exhibit isn't directly related to how big it is, but rather what level of motion you're able to measure with confidence.
Physicists at Chalmers University of Technology in Sweden have created a sort of artificial microphone on a semiconductor chip cooled to near absolute zero that's capable of measuring sound waves with an amplitude (a "volume") of just a few quintillionths of a meter. This is just a few percent of the size of a proton, which itself is significantly smaller than the size of an atom. At at these scales, classical physics starts to break down into quantum mechanics, or more specifically, "quantum acoustics," which nobody has been able to study before.
The researchers are looking for something called a "quantum phonon," which represents the smallest possible packet of sound. Even with the world's biggest amplifier, you won't be able to hear a quantum phonon when they find it: it'll have a frequency of almost one gigahertz (a billion hertz), which is way, way higher than the 20 kilohertz (20 thousand hertz or so) that's the highest frequency that humans can hear. Once the researchers can isolate and control quantum phonons, they might be useful in things like quantum computers, taking the place of light as a way of transmitting information between quantum bits.