We present results from multi-wavelength simultaneous X-ray and radio observations of the black hole X-ray binary V404 Cyg in quiescence. Our coverage with NuSTAR provides the very first opportunity to study the X-ray spectrum of V404 Cyg at energies above 10 keV. The unabsorbed broad-band (0.3--30 keV) quiescent luminosity of the source is 8.9$times$10$^{32}$ erg s$^{-1}$ for a distance of 2.4 kpc. The source shows clear variability on short time scales (an hour to a couple of hours) in radio, soft X-ray and hard X-ray bands in the form of multiple flares. The broad-band X-ray spectra obtained from XMM-Newton and NuSTAR can be characterized with a power-law model having photon index $Gamma$=2.12$pm$0.07 (90% confidence errors); however, residuals at high energies indicate spectral curvature significant at a 3$sigma$ confidence level with e-folding energy of the cutoff to be 20$^{+20}_{-7}$ keV. Such curvature can be explained using synchrotron emission from the base of a jet outflow. Radio observations using the VLA reveal that the spectral index evolves on very fast time-scales (as short as 10 min.), switching between optically thick and thin synchrotron emission, possibly due to instabilities in the compact jet or stochastic instabilities in accretion rate. We explore different scenarios to explain this very fast variability.