The axion particle, a consequence of an elegant hypothesis that resolves the strong-CP problem of quantum chromodynamics, is a plausible origin for cosmological dark matter. In searches for axionic dark matter that detect the conversion of axions to microwave photons, the quantum noise associated with microwave vacuum fluctuations will soon limit the rate at which parameter space is searched. Here we show that this noise can be partially overcome either by squeezing the quantum vacuum using recently developed Josephson parametric devices, or by using superconducting qubits to count microwave photons. The recently demonstrated ability of superconducting qubits to make QND measurements of microwave photons offers great advantages over destructive photon counting methods such as those using Rydberg atoms.