The paper investigated the experimental study of composite spherical-roof contoured-core (SRCC) sandwich panels under a V-shaped quasi-static indentation (QSI) test. The carbon fiber, glass fiber, aramid fiber, hybrid carbon/aramid, and glass fiber chopped strand mat sheets were used to manufacture the composite cores, which used the hand layup and hot compression techniques. The polylactide (PLA) core was fabricated using fused deposition modelling (FDM). The V-shaped indenter had a 65° angle and was set at 1 mm/min as indentation speed in the experimental test. It was found that the maximum indentation deflection was 35.49 mm on the SRCC sandwich panel with a hybrid carbon/aramid core compared to other composite cores. Furthermore, it was localized fiber fracture, matrix damage, delamination, inwards fronds, and debonding damage mode on composite SRCC sandwich panels, which was visually shown at the central SRCC core unit cell. It is highlighted that aramid fiber and hybrid carbon/aramid materials can provide outstanding impact characteristics, which have great potential to be used as impact-resistant sandwich structures.