A bracket mounting structure includes a bracket, a base, a connecting member, an elastic member, and a mounting member. The bracket includes a longitudinal wall. The base is arranged adjacent to the longitudinal wall and includes a first space and a second space. The connecting member is mounted in the first space and includes a first portion, a second portion, and a through hole arranged in the first portion and the second portion. At least one of the first portion and the second portion has a screw thread feature. The elastic member is received in the first space and is configured to apply an elastic force to the connecting member. The mounting member is mounted in the second space.