Semiconductors are commonly divided into materials with direct or indirect band gaps based on the relative positions of the top of the valence band and the bottom of the conduction band in crystal momentum (k) space. It has, however, been debated if k is a useful quantum number to describe the band structure in quantum-confined nanocrystalline systems, which blur the distinction between direct and indirect gap semiconductors. In bulk III-V semiconductor alloys like In1-xGaxP, the band structure can be tuned continuously from the direct- to indirect-gap by changing the value of x. The effect of strong quantum confinement on the direct-to-indirect transition in this system has yet to be established because high-quality colloidal nanocrystal samples have remained inaccessible. Herein, we report one of the first systematic studies of ternary III-V nanocrystals by utilizing an optimized molten-salt In-to-Ga cation exchange protocol to yield bright In1-xGaxP/ZnS core-shell particles with photoluminescence quantum yields exceeding 80%. We performed two-dimensional solid-state NMR studies to assess the alloy homogeneity and the extent of surface oxidation in In1-xGaxP cores. The radiative decay lifetime for In1-xGaxP/ZnS monotonically increases with higher gallium content. Transient absorption studies on In1-xGaxP/ZnS nanocrystals demonstrate signatures of direct- and indirect-like behavior based on the presence or absence, respectively, of excitonic bleach features. Atomistic electronic structure calculations based on the semi-empirical pseudopotential model are used to calculate absorption spectra and radiative lifetimes and evaluate band-edge degeneracy; the resulting calculated electronic properties are consistent with experimental observations. By studying photoluminescence characteristics at elevated temperatures, we demonstrate that a reduced lattice mismatch at the III-V/II-VI core-shell interface can enhance the thermal stability of emission. These insights establish cation exchange in molten inorganic salts as a viable synthetic route to nontoxic, high-quality In1-xGaxP/ZnS QD emitters with desirable optoelectronic properties.