Strained silicon technology employing strain-relaxed SiGe virtual substrates has become pivotal factor in advancing group IV semiconductor electronics, photonic devices, silicon-based quantum computing architectures, and neuromorphic devices. Although existing approaches using Si/SiGe superlattice buffers and compositionally graded SiGe layers can produce high-quality SiGe virtual substrates, defects including threading dislocations and crosshatch patterns still limit further performance enhancement. This study demonstrates a method of fabricating fully elastically relaxed SiGe nanomembranes that effectively suppresses the formation of both threading dislocations and crosshatch patterns. The fabrication process comprises three key steps: 1) epitaxially growing Si/SiGe/Si heterostructures on silicon-on-insulator substrates via molecular beam epitaxy (MBE), 2) fabricating periodic pore arrays by using photolithography and reactive ion etching, and 3) selectively wet etching and subsequently transferring nanomembranes to Si(001) substrates. Subsequently, a Si/SiGe heterostructure is grown on the SiGe nanomembranes via MBE. The full elastic relaxation state of the SiGe nanomembranes and the fully strained state of the Si quantum well in the epitaxial Si/SiGe heterostructures are verified using Raman spectroscopy. Surface root-mean-square roughness value is 0.323 nm for the SiGe nanomembrane transferred to the silicon substrate and 0.118 nm for the epitaxial Si/SiGe heterostructure, which are demonstrated through atomic force microscopy measurements. Through electron channel contrast imaging, it is demonstrated that the Si/SiGe heterostructures grown on SiGe nanomembranes have uniform surface contrast and no detectable threading dislocations. Comparatively, the silicon substrate region exhibits high-density threading dislocations accompanied by stacking faults. Cross-sectional transmission electron microscope analysis shows atomically sharp and defect-free interfaces. This research lays a critical foundation for developing high-mobility two-dimensional electron gas systems and high-performance quantum bits.