The decay times of D-1 emission lines of Rb and Cs atoms in superfluid helium were measured at 1.6 K as functions of helium pressure in a range from the saturated vapor pressure to about 25 atm. In the case of Cs, the measured decay time is almost independent of the helium pressure and is given by the lifetime of the 6P(1/2) state of the free atom. However, we have found that the decay time of the Rb D-1 line decreases with increasing pressure. This suggests the coexistence of two processes leading from the excited P state to the ground state, one an ordinary D-1 emission transition and the other a process decaying via a lower energy state before reaching the ground state. The pressure-dependent quenching observed only for the Rb atom can be explained by the competition of the above two processes. A possible mechanism for quenching of the Rb 5P states is discussed in terms of the interaction potentials of the alkali-metal-atom-He-N system, where two or six helium atoms are localized near the alkali core in a particular anisotropic configuration. The dynamics of processes leading from the excited P states of Na and Cs atoms are also discussed.