You could answer this question regarding conflict in two ways. You could examine this question in terms of the characterization of the protagonist, the grandmother. She is the central character in the story and the one who arguably undergoes change in the end. Her conflict is external, between herself and a society which she judges harshly. Viewing herself as a "good" person and a woman of faith, the grandmother places an emphasis on appearing to be a "lady" through her clothing. She makes racist comments about young Black children from her car, viewing them as some sort of spectacle of entertainment. She also tries to convince the Misfit that he doesn't really want to kill her, because she believes he is a "good man." She comments that she "can just look at [him] and tell." Because the grandmother places a heavy emphasis on appearances, she finds herself ultimately staring evil in the face and comes to understand in the last moments of her life that she and the Misfit are more alike than she's ever imagined.
You could also examine the conflict in terms of plot development, locating the moment in the plot which directly leads to the climax. I would say this moment occurs when the grandmother convinces the family to take the wrong road, leading to their accident and then to the Misfit finding them. Because of her faulty memory, the family is discovered and ultimately killed.
The central conflict is between the grandmother and society. She feels that people, in general, are no longer as good as they are used to be, and she bemoans this fact throughout the entire story. Her relationship with her son, Bailey, and his family is clearly not up to snuff for her: Bailey ignores her, at best, the two older children are very disrespectful to her, and the name of his wife is never even mentioned. She is unhappy with the way she is treated by the whole family.
On vacation, when the family stops at Red Sammy Butts's Barbecue, the grandmother takes the opportunity to commiserate with Red Sammy, a man who shares her opinion that "a good man is hard to find" and that it is difficult to trust anybody nowadays. No one else seems to want to engage in this kind of conversation with either of them. The irony, here, is that both the grandmother and Red Sammy are pretty terrible in their own ways: for example, she's a racist and he is very rude to his wife.
Next, the grandmother has conflict with her son again as they continue down the road, and her mistake ends up leading to the car accident that puts them in the path of the Misfit. As the Misfit's cohorts kill her family, one by one, she tries to convince him that he is a good man, that she and he are of the same ilk. He recognizes, however, that she is a part of the same strata of society that locked him up for a crime he didn't commit, ruining his life and causing him to become jaded and cruel. Ultimately, it is the grandmother's antagonism with society (in the form, first, of her son, and then with the Misfit) that leads to her death as well as her pre-death realization that she and the Misfit actually are linked, in the sense that all human beings are connected, a realization she could not have had—as the Misfit points out—without a gun pointed at her head.
The main conflict is the inner conflict of the grandmother, who mistakenly perceives herself as a good woman and superior to others. For instance, she feels it incumbent upon herself to instruct her grandchildren to be "more respectful of their native states and their parents and everything else" while she then remarks upon what a "cute little pickaninny" is standing outside the door of a shack that Bailey, her son, drives past. In another example, the grandmother tells the children a story of her youth, in which she
would have done well to marry Mr. Teagarden because he was a gentleman and had bought Coca-Cola stock when it first came out and that he had died only a few years ago....
Clearly, she feels herself deserving of such a man. In the filling station/dance hall, she talks with the proprietor, commiserating that "a good man is hard to find," implying that she is, of course, a good person herself.
However, the grandmother does admit to herself some that she is not honest. When she wants to see the plantation house, she fabricates a story about it:
She said craftily, not telling the truth but wishing that she were, "and the story went that all the family silver was hidden in it when Sherman came through but it was never found..."
Of course, her selfishness puts the family in the predicament that they find themselves after the grandmother's cat causes Bailey to lose control of the car and the Misfit and his friends appear on the scene. And, it is only at the point of a gun that the grandmother relinquishes her hypocrisy and perceives herself as a sinner, too.
"Why, you're one of my babies! You're one of my own children"
she exclaims as the Misfit stands over her, wearing her son's Bailey's shirt. Finally, as she finds redemption, the grandmother realizes that she has not led a good life and she is not superior to others and her inner conflict is resolved.