It is universally accepted that noise may bring order to complex nonequilibrium systems. Most strikingly, entirely new states not seen in the noiseless system can be induced purely by including multiplicative noise-an effect known as pure noise-induced transitions. It was first observed in superfluids in the 1980s. Recent results in complex nonequilibrium systems have also shown how new collective states emerge from such pure noise-induced transitions, such as the foraging behavior of insect colonies, and schooling in fish. Here we report such effects of noise in a quantum-mechanical system. We find that multiplicative quantum noise can induce a classically forbidden transition. We use a minimal model of a nonlinearly damped oscillator in a fluctuating environment that is analytically tractable, and whose microscopic physics can be understood. When environmental noise is included, the system is seen to transition to a limit-cycle state. The noise-induced quantum limit cycle exhibits genuine nonclassical traits such as Wigner negativity and number-parity sensitive circulation in phase space. Such quantum limit cycles are also conservative. These properties are in stark contrast to those of a widely used limit cycle in the literature, which is dissipative and loses all Wigner negativity. Our results establish the existence of a pure noise-induced transition that is nonclassical and unique to open quantum systems. They illustrate a fundamental difference between quantum and classical noise.