Magnetic energy release conversion, plasma heating and charged particle energization-acceleration in the magnetic fluctuation-induced self-generating-organization region and the plasma turbulence-induced self-feeding-sustaining region are key issues for large temporal-spatial scale turbulent magnetic reconnection (LTSTMR; observed current sheet thickness to characteristic electron length, electron Larmor radius for low-β and electron inertial length for high-β, ratios on the order of 10¹⁰ ∼ 10¹¹; observed evolution time to electron cyclotron time ratios on the order of 10¹⁰ ∼ 10¹¹) that ranges from Earth’s magnetosphere to solar eruptions and other astrophysical phenomena. As the first part of a two-paper series, this paper introduces a relativistic hybrid particle-in-cell and lattice Boltzmann (RHPIC-LBM) model that describes the continuous kinetic-dynamic-hydro fully coupled LTSTMR evolution process. First, based on the governing equations of resistive relativistic magnetohydrodynamics (MHD), the relativistic discrete distribution functions for a magnetic field (D3Q7), electric field (D3Q13), electromagnetic field (D3Q13), charged particle (D3Q19) and neutral particle (D3Q27) of different plasma species are obtained for the RHPIC-LBM lattice grid. Then, the numerical process, algorithm, pseudocode, flowchart and GPU-CPU heterogenous framework of the RHPIC-LBM are described. Finally, this model is tested and validated on Tianhe-2from the National Supercomputer Center in Guang Zhou (NSCC-GZ) with 10,000  ∼  100,000 CPU cores and 50 ∼ 120 hours per case. We investigate the solar atmosphere LTSTMR activities, including the picoscale (10E−2 m ∼10E5 m), nanoscale (10E5 m ∼10E6 m), microscale (10E6 m ∼10E7 m), macroscale (10E7 m ∼10E8 m) and large hydroscale (10E8 m ∼10E9 m). All the simulation results are consistent with observations and theories. The validated model is applied to explore the turbulence evolution of the interactions of the magnetic helical structures in the 3D LTSTMR self-generating-organization magnetic field region and self-feeding-sustaining plasma region in Part II.