We report details on the axion dark matter search experiment that uses the innovative technologies of a High-Temperature Superconducting (HTS) magnet and a Josephson Parametric Converter (JPC). An 18 T HTS solenoid magnet is developed for this experiment. The JPC is used as the first stage amplifier to achieve a near quantum-limited low-noise condition. The first dark matter axion search was performed with the 18 T axion haloscope. The scan frequency range is from 4.7789 GHz to 4.8094 GHz (30.5 MHz range). No significant signal consistent with Galactic dark matter axion is observed. Our results set the best limit of the axion-photon-photon coupling ($g_{aγγ}$) in the axion mass range of 19.764 to 19.890 $μ$eV. Using the Bayesian method, the upper bounds of $g_{aγγ}$ are set at 0.98$\times|g_{aγγ}^{\text{KSVZ}}|$ (1.11$\times|g_{aγγ}^{\text{KSVZ}}|$) in the mass ranges of 19.764 to 19.771 $μ$eV (19.863 to 19.890 $μ$eV), and at 1.76 $\times|g_{aγγ}^{\text{KSVZ}}|$ in the mass ranges of 19.772 to 19.863 $μ$eV with 90\% confidence level, respectively. We report design, construction, operation, and data analysis of the 18 T axion haloscope experiment.

PRD published version