Quantum dots (QDs), which have high color purity, are expected to be applied as emitting materials to wide-color-gamut displays. To enable their use as an alternative to Cd-based QDs, it is necessary to improve the properties of QDs composed of low-toxicity materials. Although multielement QDs such as Ag-In-Ga-S are prone to spectrally broad emission from defect sites, a core/shell structure covered with a GaSx shell is expected to enable sharp emission from band-edge transitions. Here, QD light-emitting diodes (QD-LEDs) embedded with Ag-In-Ga-S/GaSx core/shell QDs (AIGS QDs) were fabricated, and their electroluminescence (EL) was observed. The EL spectra from the AIGS QD-LEDs were found to contain a large defect-related emission component not observed in the photoluminescence (PL) spectra of the AIGS QD films. This defect-related emission was caused by electrons injected into defect sites in the QDs. Therefore, the AIGS QDs and the electron injection layer (EIL) of ZnMgO were treated with Ga compounds such as gallium chloride (GaCl3) and gallium tris(N,N'-diethyldithiocarbamate) (Ga(DDTC)3) to improve the luminescence properties of the QD-LEDs. The added Ga compounds effectively compensated for defect sites on the surface of the QDs and suppressed direct electron injection from the EIL into defect sites. As a result, the defect-related emission components in the EL were successfully suppressed, and the EL exhibited a color purity comparable to the PL of the AIGS QD films. The QD-LEDs exhibited EL spectra with a full width at half-maximum of 33 nm, which is extremely sharp for a low-toxicity QD, and the chromaticity coordinates (0.260, 0.695) for green EL were achieved.