Nanocomposite doping is an effective method to improve the dielectric properties of polyethylene. Meanwhile, the introduction of thermal conductivity groups in crosslinked polyethylene (XLPE) is also an effective way to improve the thermal conductivity. Nano-zeolite is an inorganic material with a porous structure that can be doped into polyethylene to improve the insulation performance. In this paper, hyperbranched polyarylamide (HBP) with a high thermal conductivity and an auxiliary crosslinking agent (TAIC) was grafted on the surface of ZSM-5 nano-zeolite successively to obtain functionalized nano-zeolite (TAICS-ZSM-5-HBP) (the "S" in TAICS means plural). The prepared functionalized nano-zeolite was doped in polyethylene and grafted under a thermal crosslinking reaction to prepare nanocomposites (XLPE/TAICS-ZSM-5-HBP). The structural characterization showed that the nanocomposite was successfully prepared and that the nanoparticles were uniformly dispersed in the polyethylene matrix. The space charge of the TAICS-ZSM-5-HBP 5wt% nanocomposite under a high electric field was obviously inhibited. The space charge short-circuit test showed that the porous structure of the nano-zeolite introduced more deep traps, which made the trapped charge difficult to break off, hindering the charge injection. The introduction of TAICS-ZSM-5-HBP particles can greatly improve the thermal conductivity of nanocomposites. The thermal conductivity of the XLPE/5wt% and XLPE/7wt% TAICS-ZSM-5-HBP nanocomposites increased by 42.21% and 69.59% compared to that of XLPE at 20 °C, and by 34.27% and 62.83% at 80 °C.