Through low-velocity impact testing, the effects of punch shape (conical, hemispherical, and cylindrical) and impact energy (5, 10, and 15 J) on damage characteristics in glass fiber composite pipes were investigated. Ultrasonic A-scan inspection was employed to detect internal delamination damage at the impact points within the composite pipes. Test results indicate that the contact area between the punch and the pipe is a key factor influencing the severity of pipe damage. A smaller contact area results in a higher energy absorption rate, greater punch displacement, larger area under the load-displacement curve, and longer contact time, leading to more severe damage characteristics. When the conical punch delivered 15 J of impact energy, the energy absorption rate of the glass fiber composite pipe reached 91.6%, exhibiting multiple damage characteristics, including pitting, penetration, and cross-shaped cracks. As impact energy increases, the area of internal delamination damage caused by the three punch shapes exhibits near-linear growth. The conical punch induces severe damage characteristics in the thickness direction but results in the smallest delamination area. Blunt-shaped punches (hemispherical and cylindrical) disperse impact energy over a wider region, leading to increased delamination damage area.