LuCuGaO₄ has magnetic Cu²⁺ and diamagnetic Ga³⁺ ions distributed on a triangular bilayer and is suggested to undergo a spin glass transition at T_g ∼ 0.4 K. Using μSR (muon spin rotation) and neutron scattering measurements, we show that at low temperature the spins form a short range correlated state with spin fluctuations detectable over a wide range of timescales: at 0.05 K magnetic fluctuations can be detected in both the μSR time window and also extending beyond 7 meV in the inelastic neutron scattering response, indicating magnetic fluctuations spanning timescales between ∼10^-5 and ∼10^-10 s. The dynamical susceptibility scales according to the form χ′′(ω)T^α, with α = 1, throughout the measured temperature range (0.05-50 K). These effects are associated with quantum fluctuations and some degree of structural disorder in ostensibly quite different materials, including certain heavy fermion alloys, kagome spin liquids, quantum spin glasses, and valence bond glasses. We therefore suggest that LuCuGaO₄ is an interesting model compound for the further examination of disorder and quantum magnetism.