A magnetic and thermal design of high-temperature superconducting (HTS) quadrupole magnet for newly developed superconducting proton cyclotron was presented in this research. With superconducting technology, the design can reduce the magnet size and remove the heat loads more efficiently. Calculations are conducted with finite element method (FEM) to study the quadrupole magnet’s magnetic and thermal properties. For the magnet cold-mass and cryostat system, the heat load is mainly generated by conduction and radiation. Multilayer thermal insulation and G10 supports are used to restrict them. The magnetic field distribution in the HTS coils is calculated to consider the critical current degradation of the YBCO tapes. The effects on the field gradient quality in relation to the pole tip profile and end chamfer are analyzed. The sixth multipole component in the integral field can be improved with end chamfer. Finally, a 20 T/m HTS quadrupole magnet with integral field uniformity less than 0.05% out to 75% of the inscribed radius is proposed.