We present the first comprehensive halo occupation distribution (HOD) analysis of the Dark Energy Spectroscopic Instrument (DESI) One-Percent Survey luminous red galaxy (LRG) and Quasi Stellar Object (QSO) samples. We constrain the HOD of each sample and test possible HOD extensions by fitting the redshift-space galaxy 2-point correlation functions in 0.15 < r < 32 h(-1) Mpc in a set of fiducial redshift bins. We use ABACUSSUMMIT cubic boxes at Planck 2018 cosmology as model templates and forward model galaxy clustering with the ABACUSHOD package. We achieve good fits with a standard HOD model with velocity bias, and we find no evidence for galaxy assembly bias or satellite profile modulation at the current level of statistical uncertainty. For LRGs in 0.4 < z < 0.6, we infer a satellite fraction of f(sat)=11 +/- 1 y percent, a mean halo mass of log(10)(M) over bar (h)/M-circle dot = 13.40(-0.02)(+0.02), and a linear bias of b(lin)=1.93(-0.04)(+0.06). For LRGs in 0.6 < z < 0.8, we find f(sat)=14 +/- 1 percent, log(10)(M) over bar (h)/M-circle dot =13.24(-0.02)(+0.02), and b(lin)=2.08(-0.03)(+0.03). For QSOs, we infer f(sat)=3(-2)(+8)percent, log(10)(M) over bar (h)/M-circle dot = 12.65(-0.04)(+0.09), and b(lin)=2.63(-0.26)(+0.37) in redshift range 0.8 < z < 2.1. Using these fits, we generate a large suite of high fidelity galaxy mocks, forming the basis of systematic tests for DESI Y1 cosmological analyses. We also study the redshift-evolution of the DESI LRG sample from z = 0.4 up to z = 1.1, revealling significant and interesting trends in mean halo mass, linear bias, and satellite fraction.