We use the Monte Carlo particle physics code FLUKA to calculate γ-ray spectra expected from solar flare energetic ion distributions. The FLUKA code includes robust physics-based models for electromagnetic, hadronic, and nuclear interactions, sufficiently detailed for it to be a useful tool for calculating nuclear de-excitation, positron annihilation and neutron capture line fluxes and shapes, as well as ~GeV continuum radiation from pion decay products. We show γ-ray model spectra which exhibit all the typical structures of γ-ray spectra observed in solar flares. From these model spectra we build templates which are incorporated into the software package OSPEX and used to fit the joint Fermi Gamma-ray Burst Monitor (GBM)/Large Area Telescope (LAT) spectrum of the 2010 June 12 solar flare, providing a statistically acceptable result.