I can't tell by reading your question whether you are wanting to compare the character of Emma or Austen's book called Emma with Montaigne's experience within his Essays. As a result, I am willing to approach both ideas for you. I will explore the character of Emma, then the character of Mr. Knightley (who proves to be Emma's opposite and, yet, her fulfillment), and finally Montaigne's ideas throughout his Essays. Comparison will follow.
First, let's take the character of Emma. Emma is a know-it-all who needs to be tempered by experience. Emma gets this experience through the course of Austen's novel, but only through grueling escapades within society. Emma loves to control social lives in her upper-class community, often serving as a (failed) matchmaker.
And have you never known the pleasure and triumph of a lucky guess? I pity you. I thought you cleverer; for depend upon it, a lucky guess is never merely luck. There is always some talent in it.
Emma's only real success is when Miss Taylor marries Mr. Weston, and this happens at the very beginning. It is all downhill from there for Emma's matchmaking abilities. Emma then tries to unite Harriet Smith with the minister Mr. Elton. There are a number of obstacles to the relationship; Harriet would be marrying above her station. Unfortunately, Mr. Elton has no feelings for Harriet and, worse, Harriet is in love with someone else: someone of her own class. Next Emma tries her matchmaking ability on herself and Frank Churchill. This is another failure, as Churchill is already engaged to Jane Fairfax. In a twist of fate, Emma (now upset with her failed matchmaking) finally falls in love with the one person who has always cared for her and tried to instruct her: Mr. Knightley. A marriage ensues and all live happily ever after with Emma's learning experiences being complete.
In regard to Austen's full novel Emma, it would be a good idea to discuss the character of Mr. Knightley in that he is the most similar to Montaigne. Speaking with force and direct opposition, Mr. Knightley is the only person who confronts Emma about how she meddles, unwanted, into other people's social lives. Further, Mr. Knightley admits that Emma almost always has some kind of misinformation that makes her make bad decisions.
"I cannot make speeches, Emma," he soon resumed; and in a tone of such sincere, decided, intelligible tenderness as was tolerably convincing. "If I loved you less, I might be able to talk about it more. But you know what I am. You hear nothing but truth from me."
Always reserved and proper, Mr. Knightley allows Emma to learn through her own errors, even when her learning experiences prove embarrassing.
About Montaigne himself, one must consider that the actual meaning of the word essay is "test" or "try." I remember my high school English teacher saying that every essay was "an attempt," which was directly from Montaigne. Let's put Montaigne on a pedestal here, where he belongs, and admit that the man actually INVENTED the essay that so many students write today. I love the way eNotes states Montaigne's methods to teach (and to have) learning experiences:
Even more than Socrates, he believed that the awareness of one’s own ignorance is the basis of wisdom. Instead of insisting on the correctness of his ideas, he attempts to see his subjects from other points of view, including those of Mohammedans, cannibals, and even of cats.
Montaigne values actual experiences as much as learning from books. Thus, Montaigne allows us to have our own learning experiences while we consider his words and ideas. He does all this with humor and with informality that allow the reader to become even more enamored with learning from him.
In conclusion, Emma takes the opposite approach from that of both Mr. Knightly and Montaigne. You can see how Emma has learning experiences during Austen's novel that reveal her to make poor decisions while she thinks the exact opposite of herself. This know-it-all quality is tempered over the course of the book by the remonstrations gently given by Mr. Knightly who always suggests that Emma not be so sure about her own ideas. Mr. Knightly, then, compares quite nicely with Montaigne himself who "tests" and "tries" in his different essays that say less about what he himself knows and more to encourage the reader to figure out what he/she knows and feels for himself/herself. To Montaigne, we are the "Emmas" of the world, learning through his Mr. Knightley-esque instruction.