Does The hydrogen bomb rely on the process of nuclear fission.

Yes it does.

Nuclear reactions have two main types: fission reactions and fusion reactions. When two light nuclei are joined to create a heavier and more stable nucleus, this process is called the fusion reaction. In the fission reaction, the heavy nucleus split into two nuclei with a smaller mass number. In other words, fission is splitting of atoms although not all atoms undergo fission, only few went through normal circumstances (that is a fact).