Inimitable properties of graphene sheets enable a variety of applications such as axially moving nanodevices. Axial velocity affects dynamical response of systems. In this study linear vibration of an axially moving two-layer graphene nonoribbon with interlayer shear effect is proposed using nonlocal elasticity theory. Based on this theory stress at a point is a function of strain at all other points of the body. Euler-Bernoulli theory is used to model the system due to nanoribbon thickness and length. It is assumed that the layers have the same transverse displacement and curvature and there is no transverse separation between layers surfaces. A shear modulus is imported in the potential energy expression in order to consider the interlayer shear effect due to weak Van der Waals forces. Governing equations are obtained using Hamilton’s principle and are solved by Galerkin approach. Results for clamped-free boundary conditions are presented and compared to other available studies. Results for pinned-pinned boundary conditions are presented and it is observed that increasing axial velocity causes divergence and flutter instabilities in the system. Effects of different shear modulus and nonlocal parameter on critical speeds are also proposed.