Ultra–high quality factor nanomechanical resonators have traditionally relied on periodic phononic crystal architectures, where bandgap-enabled soft clamping isolates mechanical motion from the substrate. While effective, such periodic designs impose intrinsic symmetry and unit-cell constraints that limit the accessible design space. In this contribution, we investigate quasicrystal-based architectures as a non-periodic alternative for nanomechanical resonator design, emphasizing their potential to expand structural freedom while remaining compatible with soft-clamping concepts. We introduce a data-driven design workflow that enables the systematic exploration of quasicrystal geometries without relying on translational periodicity or conventional band-structure analysis. By combining symmetry-aware numerical analysis with automated simulation and optimization tools, the framework allows efficient identification of mechanically isolated modes and quantitative evaluation of resonator performance across complex, aperiodic geometries. Using representative quasicrystal-inspired membrane resonators, we demonstrate that aperiodic architectures can support strongly localized vibrational modes with performance comparable to established periodic designs. Beyond serving as an alternative structural motif, quasicrystal geometries naturally introduce additional design flexibility in mode localization and stress distribution that is difficult to achieve within periodic lattices. These results position quasicrystals as a promising platform for next-generation nanomechanical resonators and highlight the role of data-driven approaches in translating complex structural order into functional mechanical performance. The presented framework points toward resonator design strategies that extend beyond periodic order, with relevance to precision sensing and emerging quantum-enabled mechanical systems.