Control of composition, stoichiometry, and defects in colloidal quantum dots (QDs) of III-V semiconductors has proven to be difficult due to their covalent character. Whereas the synthesis of colloidal indium pnictides such as InP, InAs, and InSb has made significant progress, gallium-containing colloidal III-V QDs still remain largely elusive. Gallium pnictides represent an important class of semiconductors due to their excellent optoelectronic properties in the bulk; however, the difficulty with the synthesis of gallium containing colloidal III-V QDs has largely prohibited their exploration as solution-processed semiconductors. Here we introduce molten inorganic salts as high-temperature solvents for the synthesis and manipulation of III-V QDs. We demonstrate cation exchange reactions on presynthesized InP and InAs QDs to form In1-xGaxP and In1-xGaxAs QDs at temperatures above 380 degrees C. This approach produces novel ternary alloy QDs with controllable compositions that show size- and composition-dependent absorption and emission features. Emission quantum yields of up to similar to 50% can be obtained for In1-xGaxP/ZnS core-shell QDs. A comparison of the optical properties of InP/ZnS core-shells with In1-xGaxP/ZnS core-shells reveals that Ga incorporation leads to significant improvement in the optical properties of III-V/II-VI core-shell emitters which is of great importance for quantum dot-based lighting and display applications. This work also demonstrates the potential of molten inorganic salts as versatile solvents for the synthesis and processing of colloidal nanomaterials at temperatures inaccessible for traditional solvents.