Could population III stars develop directly into population I stars? Hundreds of millions of years after the Big Bang, the very first stars began to form consisting of mostly hydrogen, a bit of helium, and maybe some lithium. These stars lacked any "metals" (elements heavier than helium) and are categorized as Population III stars. We have yet to observe any of these Pop III stars, and observation remains unlikely. The reason for this is because Pop III stars are thought to have been incredibly massive, so they died out so long ago that we can no longer see their light. Pop I stars are metal-rich; their metallicity is 1/10 to 3 times that of our Sun's. In my research, everyone seems to agree that that the first generation of stars had no metal, second generation stars had very little, and third generation stars have some metal.But, no one ever addresses the possibilities. So, is it possible that some Pop III stars were so massive that they were able to skip Pop II and go straight to Pop I? Maybe if several are close by when they go supernova?
No they couldn't. Population I stars contain elements like strontium, barium, gold, lead etc. that are not formed (much) in type II (core collapse) supernovae. They are also much richer in iron, nickel, manganese etc. than are stars formed from gas enriched only by supernovae.

Population I stars (like the Sun) can be termed third generation stars because they contain material that has been through at least two stars - but not just two high mass stars that exploded as supernovae.

Iron-peak elements are mainly formed and disseminated by type Ia supernova that are exploding white dwarfs. White dwarfs are the remnants of low-mass stars with long lives.

Many heavy elements (strontium, barium, lead etc) are formed by the s-process in intermediate mass stars that also have (relatively) long lives and do not explode as supernovae. Furthermore, this neutron capture requires iron-peak nuclei to act as seeds, so these stars must in turn have formed out of material already enriched with iron-peak nuclei.

It is also now thought that other elements (silver, gold, osmium etc) are formed by the merger of neutron stars, where there must be a significant delay between the initial supernovae that produced the neutron stars and the subsequent merger by orbital decay.