An analytical and meshless modeling for solving thickness shear vibration in an infinite piezoelectric quartz resonator with partial non-circular electrodes is proposed. Firstly, two-dimensional scalar differential equations derived by Tiersten and Smythe are adopted and transformed into the polar coordinate system. Secondly, displacement patterns for the electroded and unelectroded regions are assumed as a series of converging and outgoing cylindrical waves in the form of Bessel functions, where radiation conditions at infinity can be satisfied automatically. Finally, circumferential functions at interface are decomposed into Fourier series in order to deal with continuity conditions. It should be stressed that the general formulation proposed in this paper has a higher calculation accuracy and requires no division of the mesh compared to FEM/BEM, which can satisfy continuity conditions in an integrated manner over the whole interface or boundary. Resonance frequencies and mode shapes of different electrode shapes including circular, equilateral triangle, rectangular, elliptical, and pentagonal electrodes are numerically calculated and compared with FEM simulations, which efficiently validate high precision and wide applicability of this method. Utilizing this method, the influence of non-circular electrodes on the working performance of the quartz resonator is investigated systematically. The qualitative analysis and quantitative results obtained in this paper can provide the theoretical guidance for the design, measurement and manufacturing optimization of piezoelectric resonators.