During the Fourth Shinobi World War, it is revealed that after losing to Hashirama at the Valley of the End, Madara implanted Hashirama's cells into his wounds
which led to him awakening the Rinnegan.
Later, it is also revealed that:
Madara and Hashirama are transmigrants of Indra and Ashura, and one needs both Indra's and Ashura's chakra to awaken the Rinnegan.
If Hashirama had instead implanted Madara's cells into his body, would he also have
awakened the Rinnegan?
It's not the cells but the eyes he would need to have Rinnengan, this is from Naruto wiki:
Because the Rinnegan was wielded by the Sage of the Six Paths, merging the chakra of his two sons, Indra Ōtsutsuki and Asura Ōtsutsuki, within one's body brings forth the Sage's chakra, which can evolve their Sharingan into the Rinnegan. This was proven when Madara Uchiha, Indra's reincarnation at the time, awakened the dōjutsu from his Eternal Mangekyō Sharingan by infusing the DNA of Hashirama Senju, Asura's reincarnation at the time, into his body; however, it did not manifest until many years later when he was near death. At some point after this, Madara would implant his eyes into Nagato — a young boy of Senju lineage.
Also there is a theory from Orochimaru and Kabuto, that says Sharingan came from Rinnegan, so without these eyes no Rinnegan:
They theorised that because the Sharingan originated from the Rinnegan, it was possible for the dōjutsu to change into the Rinnegan as part of a "natural evolution"
