We explore X-ray evidence for the presence of active galactic nuclei (AGNs) in the two most actively star-forming Green Pea galaxies (GPs), SDSS J0749+3337 and SDSS J0822+2241, which have star-formation rates (SFRs) of 123 M_sun yr^-1 and 78 M_sun yr^-1, respectively. The GPs have red mid-infrared (MIR) spectral energy distributions and higher 22 um luminosities than expected from a proxy of the SFR (Halpha luminosity), consistent with hosting AGNs with 2--10 keV luminosities of ~10^43-44 erg s^-1. We thus obtain and analyze the first hard (> 10 keV) X-ray data observed with NuSTAR and archival XMM-Newton data below 10 keV. From the NuSTAR ~20 ksec data, however, we find no significant hard X-ray emission. By contrast, soft X-ray emission with 0.5--8 keV luminosities of ~10^42 erg s^-1 is significantly detected in both targets, which can be explained only by star formation (SF). A possible reason for the lack of clear evidence is that a putative AGN torus absorbs most of the X-ray emission. Applying a smooth-density AGN torus model, we determine minimum hydrogen column densities along the equatorial plane (N_H^eq) consistent with the non-detection. The results indicate N_H^eq > 2x10^23 cm^-2 for SDSS J0749+3337 and N_H^eq > 1x10^24 cm^-2 for SDSS J0822+2241. Therefore, the GPs may host such heavily obscured AGNs. Otherwise, no AGN exists and the MIR emission is ascribed to SF. Active SF in low-mass galaxies is indeed suggested to reproduce red MIR colors. This would imply that diagnostics based on MIR photometry data alone may misidentify such galaxies as AGNs.