No, the possessive apostrophe doesn’t apply to possessive pronouns. Apostrophes in pronouns are for abbreviations (like “he is” becomes “he’s” and “it is” becomes “it’s”)
https://dictionary.cambridge.org/grammar/british-grammar/pronouns-possessive-my-mine-your-yours-etc
And: https://promova.com/content/86_Possessive_Pronouns_in_English_41a7b3d23c.png