Underwater gliders are well known as a type of autonomous observation platform of ocean phenomena, which can achieve space motion by low energy consumption. In this paper, we study motion trajectory design methods of underwater gliders executing underwater fixed-point exploration missions. First, performance analysis models of an underwater glider are established, including the dynamic model and the energy consumption model. Then, according to the positions of target exploration points, we divide underwater fixed-point exploration missions into three categories, namely, the short voyage range exploration mission, the moderate voyage range exploration mission, and the long voyage range exploration mission. Based on the glider motion modes of in-profile gliding and spiral gliding, two trajectory design methods are proposed for the above three categories of exploration missions. To achieve the full coverage of target exploration points, we present a net buoyancy compensation strategy of the glider carrying out the spiral motion. Based on the surrogate model technology and a multi-objective optimization algorithm, the optimization method of the glider control parameters is further proposed to design the optimal motion trajectory. Finally, the trajectory design methods are validated by numerical examples. This study can provide certain guidance for the actual application of underwater gliders.