Galaxy clusters are the most massive gravitationally-bound structures in the universe. The number density of clusters as a function of mass can be used to precisely quantify dark matter and other cosmological parameters. When the hot gas between galaxies in a cluster is in hydrostatic equilibrium, its gravitational potential energy is thermalized, meaning its mass can be estimated from the temperature and density of the gas or intracluster medium (ICM). X-ray telescopes are ideal for measuring this temperature via the bremsstrahlung emission spectrum of the ICM. Unfortunately, mass estimates have a systematic uncertainty due to discrepancies in temperature measurements of the same clusters by two high capability X-ray observatories, Chandra and XMM-Newton. However, the X-ray observatory NuSTAR has a greater sensitivity above the exponential turnover in the spectrum resulting in more precise and potentially more accurate galaxy cluster temperature estimates. I am focusing on measuring the temperatures of a sample of relaxed clusters, which are more likely to be in hydrostatic equilibrium and have minimal temperature variations, with NuSTAR data that already have good measurements from both Chandra and XMM-Newton. For the NuSTAR analysis, cluster spectra are modeled with a single temperature and fit in the hard energy band (3-20 keV). To prepare the data, light curves are used to filter out periods of high background. The background is characterized to allow it to be accurately subtracted from the cluster spectra. In the cool core clusters, spectra are extracted and fit in two regions to account for the cross-contamination of the cool core emission into the measurement region due to NuSTAR's large point spread function. Additionally, NuSTAR has strong precision which I use to constrain both the NuSTAR-Chandra and NuSTAR-XMM-Newton temperature relations for clusters with temperatures in the 5-10 keV range. With the results, we hope to better understand the temperature discrepancy in X-ray astronomy and improve cosmological constraints derived using galaxy clusters.