In this paper, a nonlocal foundation model is proposed to analyze the vibration and instability of a Y-shaped single-walled carbon nanotube (Y-SWCNT) conveying fluid. In order to achieve more accurate results, fourth order beam theory is utilized to obtain strain-displacement relations. For the first time, a nonlocal model is presented based on nonlocal elasticity and the effects of nonlocal forces from adjacent and non-adjacent elements on deflection are considered. The Eringen’s theory is utilized due to its capability to consider the size effect. Based on Hamilton’s principle, motion equations as well as boundary conditions are derived and solved by means of hybrid analytical-numerical method. It is believed that the presented general foundation model offers an exact and effective new approach to investigate vibration characteristics of this kind of structures embedded in an elastic medium. The results of this investigation may provide a useful reference in controlling systems in nano-scale.