We report the first search results for axion dark matter using an 18 T high-temperature superconducting magnet haloscope. The scan frequency ranges from 4.7789 to 4.8094 GHz. No significant signal consistent with the Galactic halo dark matter axion is observed. The results set the best upper bound of axion-photon-photon coupling (g_{aγγ}) in the mass ranges of 19.764 to 19.771 μeV (19.863 to 19.890 μeV) at 1.5×|g_{aγγ}^{KSVZ}| (1.7×|g_{aγγ}^{KSVZ}|), and 19.772 to 19.863 μeV at 2.7×|g_{aγγ}^{KSVZ}| with 90% confidence level, respectively. This remarkable sensitivity in the high mass region of dark matter axion is achieved by using the strongest magnetic field among the existing haloscope experiments and realizing a low-noise amplification of microwave signals using a Josephson parametric converter.