Discovery of the HArF molecule associated with H-Ar covalent bonding [Nature, 2000, 406, 874-876] has revolutionized the field of noble gas chemistry. In general, this class of noble gas compound involving conventional chemical bonds exists as closed-shell species in a singlet electronic state. For the first time, in a bid to predict neutral noble gas chemical compounds in their triplet electronic state, we have carried out a systematic investigation of xenon inserted FN and FP species by using quantum chemical calculations with density functional theory and various post-Hartree-Fock-based correlated methods, including the multireference configuration interaction technique. The FXeP and FXeN species are predicted to be stable by all the computational methods employed in the present work, such as density functional theory (DFT), second-order Møller-Plesset perturbation theory (MP2), coupled-cluster theory (CCSD(T)), and multireference configuration interaction (MRCI). For the purpose of comparison we have also included the Kr-inserted compounds of FN and FP species. Geometrical parameters, dissociation energies, transition-state barrier heights, atomic charge distributions, vibrational frequency data, and atoms-in-molecules properties clearly indicate that it is possible to experimentally realize the most stable state of FXeP and FXeN molecules, which is triplet in nature, through the matrix isolation technique under cryogenic conditions.