I would add to the first answer that another reason for the conflict is Mrs. Merriweather's reference to Atticus helping Tom Robinson; it is clear that she does not approve of it. After she complains to the other women that "some people" think they're doing the right thing by defending Tom, Miss Maudie snaps, and I paraphrase here: "His food doesn't stick going down, does it?"
This is Maudie's way of showing her disapproval of Mrs. Merriweather's absurd, racist remarks, and especially her attempt to criticize Atticus (while eating his food!)
The answer you are looking for is in Chapter 24.
This is after Tom Robinson's trial has finished. Miss Maudie and Mrs. Merriweather and some other ladies are at the Finches' home for the Missionary Circle meeting. This is where the conflict comes in.
Miss Merriweather and the others are saying horrible things about black people in general and about specific black people as well. Miss Merriweather in particular is complaining about her cook and how she is moping around because of the Robinson verdict. Miss Maudie thinks that she is complaining to much and that she should have more understanding. This is why she kind of snaps at Mrs. Merriweather.
In Chapter 24, Aunt Alexandra hosts the missionary circle at the Finch home. During the social event, Mrs. Merriweather begins to discuss the current events that are happening in Maycomb and indirectly comments on Atticus's defense of Tom Robinson. Mrs. Merriweather says that there are some "good but misguided people in this town" (Lee 142). She disagrees with Atticus's decision to defend a black man and indirectly says that Atticus is causing a disruption throughout the town. Miss Maudie recognizes Mrs. Merriweather's passive criticism of Atticus and defends him by saying,
"His food doesn't stick going down, does it?" (Lee 142)
Miss Maudie subtly brings attention to the fact that Mrs. Merriweather is criticizing a man whose home she is presently socializing in. Mrs. Merriweather's face turns red, and she quickly changes the subject. However, Aunt Alexandra appreciates Miss Maudie's defending her brother in front of the women and subtly smiles at Maudie.