The answer to this question will depend on where we believe the climax takes place.
We might argue that the climax of this book occurs when Lennie kills Curley's wife. This is the moment to which the book has been building. Lennie has snuck away to be alone and has killed (again, like he did in Weed) without meaning to hurt anyone. The latent violence in Lennie's character emerges with horrific results.
This scene is presents the greatest peak of tension in the story and fulfills several predicted/foreshadowed outcomes. We see here the reason behind George's worry that Curley's wife would cause trouble for Lennie and we see as well the darker side of Lennie's innocence, which is a complete lack of self-restraint or self-control.
When he does touch her hair he is too rough, and she tells him to stop, but he continues to clutch it.
If we view this scene as the climax, the falling action of the story consists of the brief hunt for Lennie and George's act. We can say that George shooting Lennie is the effective conclusion of the book.
However, if you prefer to identify the climax of the story as the moment when George shoots Lennie, there is very little falling action. George admits to Slim that Lennie had killed a woman in Weed and the two walk off.