The elastic buckling and vibration of quadrilateral single-layered graphene sheets (SLGS) and multi-layered graphene sheets (MLGS) embedded in polymer matrix are studied employing nonlocal continuum mechanics. Although leads to the over predicting results. In order to capture the small scale effects in nonlocal continuum theory it is assumed that the stress at a point depends on the strain at all points in the domain. This is contrary to the classical (local) continuum theory in which it is assumed that the stress at a point is just a function of the strain at that point. So, the nonlocal theory contains information about long range interactions between atoms and internal scale length. Besides the nonlocal continuum theory based models are physically reasonable from the atomistic viewpoint of lattice dynamics and molecular dynamics (MD) simulations. Small scale effects are taken into consideration. The principle of virtual work is employed to derive the governing equations. The Galerkin method in conjunction with the natural coordinates of the nanoplate is used as a basis for the analysis. The straight-sided quadrilateral domain is mapped into a square domain in the computational space using a four-node element. The discretizing and programming procedures are straightforward and easy. The non-dimensional buckling load and natural frequency of skew, rhombic, trapezoidal and rectangular nanoplates considering various geometrical parameters are obtained and for each case the effects of the small length scale are investigated. For MLGS embedded in polymer matrix the dependence of small scale effect on thickness, elastic modulus, polymer matrix stiffness and interaction coefficient between two adjacent sheets is illustrated. It is clearly observed that number of layer effect is negligible for lower (first, two and three) modes while it is significant for higher modes. Further it is found that decreasing effect of number of layer on small scale effect continues to specific number of layer (e.g. five layers for fifth mode and seven layers for seventh mode) and then frequency ratio converges to value of it for lower modes. It is shown that nonlocal effects are very important in arbitrary quadrilateral graphene sheets and their inclusion results in smaller buckling loads. Also the effects of geometrical parameters such as aspect ratio, angle and mode number on the buckling load decrease when scale coefficient increases, for all arbitrary quadrilateral graphene sheets. This phenomenon is attributed to the size effects.keywordes: small scale, buckling, vibration, quadrilateral graphine sheet, nonlocal elasticity, Galerkin method