The interface heat transfer of two layers induced by van der Waals (vdW) contacts is theoretically investigated, based on first-principles calculations at low temperatures. The results suggest that out-of-plane acoustic phonons with low frequencies dominate the interface thermal transport due to the vdW interaction. The interface thermal conductivity is proportional to the cubic of temperature at very low temperatures, but becomes linearly proportional to temperature as temperature increases. We show that manipulating the strain alters vdW coupling, leading to increased interfacial thermal conductivity at the interface. Our findings provide valuable insights into the interface heat transport in vdW heterostructures and support further design and optimization of electronic and optoelectronic nanodevices based on vdW contacts.