The climax of a story is defined as the turning point of the plot, or the moment of highest intensity.
In Muriel Barbery's The Elegance of the Hedgehog the moment of highest intensity occurs when Renee dies after barely having discovered her personal meaning of life. However, it is arguable that prior to her death, another high intensity moment was her first encounter with Paloma. As a result of that encounter, Renee's life begins to change. She meets Kakuro Ozu via Paloma, and she begins to realize that life is worth living. It is because Paloma that Renee finds a reason to live, and how she begins to slowly unfold into intellectual and emotional freedom.
Unfortunately, her taste of freedom is short-lived. Renee dies, but leaves her legacy onto Paloma who, in turn, also finds a reason to live thanks to Renee.
In conclusion, the first meeting between Paloma and Renee definitely becomes the catalyst for change in the lives of the women, and Renee dies after finally experiencing the true meaning of life.