The Hermite-Gaussian (HG) beam has many potential advanced applications in optical communications, electron acceleration, nonlinear optics and bio-optical disease detection, owing to its distinctive mode and intensity distribution. The research on HG beam are significant in the development of optics, medicine and quantum technology. However, the controlling of the evolutions of HG beam with quadratic phase modulation (QPM) in fractional systems under variable coefficients and potentials has been rarely studied. In this work, the propagation dynamic behaviors of the HG beam with QPM are investigated based on the fractional Schrödinger equation (FSE) under different variable coefficients and potentials by using a split-step Fourier algorithm. In the free space, the focusing spot of the beam becomes larger as the positive QPM coefficient increases or the Lévy index decreases. The QPM coefficient has little effect on the focusing amplitude when the Lévy index is 2. When the QPM coefficient is negative, the focusing of the beam disappears. Under the joint action of cosine modulations and QPM, the transmission of the beam oscillates not by the cosine law, but presents a large and a small breathing structure. The positive and the negative coefficient of QPM only alter the breathing sequence. The evolution period and width of the beam decrease as the modulation frequency increases. The trajectory of split beams turns into a parabolic shape under the linear modulation. In the joint influence of linear modulations and QPM, the HG beam exhibits either focusing or not focusing. Furthermore, the focusing position and focal plane of the beam decrease as the Lévy index increases. When the Lévy index is small, the beam keeps a straight-line transmission without distortion at a longer distance under the joint effect of the power function modulation and a positive QPM. The transmission of the beam also stabilizes and the beam width becomes larger with a negative QPM. Under a linear potential, the splitting of the HG beam disappears with the increase of the linear coefficient and shows a periodic evolution. The propagation trajectory of the beam shows a serrated pattern. By adding QPM, the beam is significantly amplified. Additionally, the evolution period of the beam is inversely proportional to the linear coefficient, and the transverse amplitude turns larger as the Lévy index increases. The interference among beams is strong, but it also exhibits an autofocus-defocusing effect under the joint action of a parabolic potential and QPM. In addition, the positive coefficient and the negative coefficient of QPM only affect the focusing time of the beam. The frequency of focusing increases as the Lévy index and parabolic coefficient rise. These features are important for applications in optical manipulations and optical focusing.