At what moment does Elizabeth begin to fall in love with Mr. Darcy?
In Pride and Prejudice, Elizabeth's love for Darcy begins to develop when she visits Pemberley in Chapter 43. Remember that by this point in the novel, Elizabeth knows that Mr. Wickham's stories about Darcy are all false and, as such, she is ready to change her perception of him accordingly. This process begins when Elizabeth meets with the housekeeper of Pemberley and learns about Darcy's true, and very gentle, nature:
"If I was to go through the world, I could not meet with a better. But I have always observed that they who are good-natured when children are good-natured when they grow up; and he was always the sweetest-tempered, most generous-hearted, boy in the world.''
When Darcy unexpectedly returns to the house, this process is continued. Darcy speaks and behaves just as his housekeeper has described him, which comes as some shock to Elizabeth:
"Never in her life had she seen his manners so little dignified, never had he spoken with such gentleness as on this unexpected meeting. "
This change in his manner prompts Elizabeth to realize that she may have misjudged him in the past. This sets the scene for their future, more positive, encounters and, more importantly, begins to warm Elizabeth's heart towards him.
I would say there isn't a single moment. Elizabeth's love for Darcy sneaks up on her as her opinion of him changes. There are many events that gradually change her opinion. First, Darcy's letter begins to make her wonder if she has misjudged him. Her visit to his house changes her opinion more - first from hearing how kind and humble he is from the servant, and then in seeing him when he shows up unexpectedly. Eventually, Elizabeth discovers that Darcy is the one who paid all of Wickham's debts and caused him to marry Lydia. All of these things reshape Elizabeth's knowledge of Darcy's character and cause her to gradually fall in love with him.