The heat generation in the high-$T_c$ superconducting (HTS) wire is related with the cost efficiency and safe factor of HTS devices. This paper deals with the thermal quench at the conduction-cooled joint between HTS wire and copper terminals. The 3-D numerical simulation of thermal distributions in part of the copper terminals was implemented and the premature quench at copper block was observed through the test. The results will be helpful to design the conduction-cooled HTS magnets.