Graphene, a two-dimensional (2D) single layer of carbon atoms, has attracted tremen- dous attention because of its unique electronic properties[1] and potential applications in electronic devices[2]. Earlier studies have focused on characterizing the unusual electronic and transport properties of graphene, particularly as a massless Dirac fermion system[1, 2]. More recently, some attention has been shifted to the structural stability of graphene[2–5], which is critically important to realizing the potential applications of graphene. On the one hand, as a 2D membrane structure, graphene provides an ideal testing ground[3, 4] for the classical Mermin-Wagner theorem on the existence of long-range crystalline order in 2D[6, 7]. On the other hand, the free edges of graphene are amenable to edge instabilities[5, 8–10]. The graphene edge stability is characterized by two fundamental thermodynamic quan- tities: edge energy and edge stress. The edge of a 2D structure can be understood in analogy to the surface of a 3D structure[11, 12]: the edge (surface) energy accounting for the energy cost to create an edge (surface) defines the edge (surface) chemical stability; the edge (surface) stress accounting for the energy cost to deform an edge (surface) defines the edge (surface) mechanical stability. First-principles calculations showed that chemically the armchair edge is more stable with a lower energy, while the zigzag edge is metastable against reconstruction[8], and both edges are hydrogenated in an H-rich environment[9]. Empirical-potential calculations showed that both intrinsic edges are under compressive stress rendering a mechanical edge twisting and warping instability[10]. Usually, stress and mechanical instability are understood as phenomena of classical me- chanics, but they are expected to be affected by quantum effects which become prominent at nanoscale. So far, however, quantum effects have been mostly shown for electronic struc- ture and energetic quantities of low-dimensional nanostructures. Here, we demonstrate an interesting example of quantum manifestations of mechanical quantities in graphene edge stress. Using first-principles calculations, we predict that the armchair edge stress in a nanoribbon exhibits a large oscillation with ribbon width arising from quantum size effect, while the zigzag edge stress is reduced by spin polarization. Such quantum effects on edge stress in turn manifest in graphene edge mechanical instability, with ”quantum” features that apparently cannot be described by empirical force-field potentials or continuum theory. Our calculations were performed using the method based on density functional theory (DFT) in the generalized gradient approximation, with the Perdew-Burke-Ernzerhof func- tional for electron exchange and correlation potetnials, as implemented in the VASP code 1