Axion-like particles (ALPs) are light, pseudoscalar particles that are a beyond-the-standard-model generalization of the axion. Consequently, they are expected to couple to photons in external magnetic fields to compensate for spin difference. This coupling would induce modifications to the gamma-ray spectra of astrophysical sources, such as blazars, via ALP-photon oscillations in external fields near the source and in the Galactic magnetic field. In this contribution, we explore ALP-photon oscillation effects in the spectrum of the blazar Markarian 421. This work was performed using observations of an exceptional gamma-ray flare state from VERITAS and Fermi-LAT. Using these observations, we investigate constraints on the two parameters defining the ALP, namely its mass and coupling constant.