The propagation dynamics of Gaussian beams and finite energy Airy beams with spectral quadratic phase modulation (QPM) modeled by the fractional Schrödinger equation (FSE) are numerically investigated. Compared with beam propagation in the standard Schrödinger equation, the focusing property of beams under FSE is influenced by the QPM coefficient and the Lévy index. For symmetric Gaussian beams, the focusing position increases and the focusing intensity decreases for the larger QPM coefficient or smaller Lévy index. For asymmetric Airy beams, multiple focusing positions occur, and the tendency of focusing intensity is opposite to that of Gaussian beams. Our results show the promising application of the FSE system for optical manipulation and optical splitting by controlling the QPM.