Quasicrystals remain among the most intriguing materials in physics and chemistry. Their structure results in many unusual properties, including anomalously low friction as well as poor electrical and thermal conductivity, but it also supports superconductivity, which shows that quantum effects in quasicrystals can be quite unique. We theoretically study superfluidity in a model quantum cluster quasicrystal. Using path-integral Monte Carlo simulations, we explore a two-dimensional ensemble of bosons with the Lifshitz-Petrich-Gaussian pair potential, finding that moderate quantum fluctuations do not destroy the dodecagonal quasicrystalline order. This quasicrystal is characterized by a small yet finite superfluidity, demonstrating that particle clustering combined with the local cogwheel structure can underpin superfluidity even in the almost classical regime. This type of distributed superfluidity may also be expected in certain open crystalline lattices. Large quantum fluctuations are shown to induce transitions to cluster solids, supersolids, and superfluids, which we characterize fully quantum mechanically.