  Non-thermal emission from clusters of galaxies at the high-energy X-ray
regime has been searched with various instruments, but the detection
significance of this emission has yet been found to be either marginal or
controversial. Taking advantage of NuSTAR's unique capability to focus X-rays
in the hard energy band, we present a detailed analysis of 238 ks NuSTAR
observations of the merging galaxy cluster SPT-CL J2031-4037, searching for
non-thermal inverse Compton emission. Our spectral analysis of SPT-CL
J2031-4037 shows a possibility that the hard X-ray emission of the cluster can
be described by a non-thermal component, though we cannot completely rule out a
purely thermal origin for this hard emission. Including the statistical and
systematic uncertainties, our best model fit yields a 20-80 keV non-thermal
flux of $3.93_{-1.10}^{+1.24} \times 10^{-12}$ erg s$^{-1}$ cm$^{-2}$. The
estimated non-thermal flux is comparable to those found in other galaxy
clusters using NuSTAR and other X-ray instruments. Using this non-thermal flux
with the existing radio data of the cluster, we estimate a volume-averaged
magnetic field strength in the range of around 0.1-0.2 $\mu$G.
