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Nuclear quantum effects are essential for an accurate description of the hydrogen-bonded structure of liquid water, yet their spectroscopic signatures can be subtle and difficult to measure. Moreover, probing these effects in solution on femtosecond timescales (or below) is an open challenge that is pivotal for optical control of hydrogen bonds and ultrafast spectroscopy of proton transfer. Here, we combine path-integral molecular dynamics with real-time time-dependent density-functional theory to explore high-harmonic generation (HHG) in liquid water from first principles, rigorously account- ing for quantum nuclear delocalisation. Relative to classical-nuclei simulations, quantum sampling suppresses the harmonic yield around certain harmonics by up to an order of magnitude, inducing local structural minima. By analysing the ab initio simulations and reproducing the key effects with a simplified model Hamiltonian, we trace the physical origins of the interference minima to the broadening of valence-band states induced by hydrogen-bond fluctuations, a mechanism expected to generalise to other hydrogen-bonded structures. Our work shows that nuclear quantum effects are indispensable for capturing ultrafast electronic dynamics in liquids, and demonstrates the power of liquid HHG spectroscopy for attosecond probing of nuclear fluctuations.