Detection of the quantum fluctuations by conventional methods meets certain obstacles, since it requires high-frequency measurements. Moreover, quantum fluctuations are normally dominated by classical noise and are usually further obstructed by various accompanying effects such as a detector backaction. In the present work, we demonstrate that these difficulties can be bypassed by performing the cross-correlation measurements. We propose to use a pair of two-level detectors, weakly coupled to a collective mode of an electric circuit. Fluctuations of the current source accumulated in the collective mode induce stochastic transitions in the detectors. These transitions are then read off by quantum-point contact (QPC) electrometers and translated into two telegraph processes in the QPC currents. Since both detectors interact with the same collective mode, this leads to a certain fraction of the correlated transitions. These correlated transitions are fingerprinted in the cross correlations of the telegraph processes, which can be detected at zero frequency, i.e., with long-time measurements. Concerning the dependance of the cross correlator on the detectors' energy splittings ɛ1 and ɛ2, the most interesting region is at the degeneracy points ɛ1=±ɛ2, where it exhibits a sharp nonlocal resonance, that stems from higher-order processes. We find that at certain conditions, the main contribution to this resonance comes from the quantum noise. Namely, while the resonance line shape is weakly broadened by the classical noise, the height of the peak is directly proportional to the square of the quantum component of the noise spectral function.