This poem does not contain a direct conflict. Rather, it contains a situation that obviously comes after some form of conflict, where the man who is eating his breakfast is choosing not to talk to the speaker of the poem.
The poem describes a typical breakfast scene in painstaking detail, including the making of a cup of coffee, the smoking of a cigarette, and the donning of a hat and raincoat. All of these are done without a word being said between the two characters, which creates an obvious atmosphere of tension throughout the poem, which culminates in the narrator's tears after the man finishes his breakfast and leaves.
While we are not given any information as to the nature of the conflict that has taken place, the assumption is that it is between a husband and a wife. It is worth noting that, while the narrator is devastated by what has happened, she makes no effort to talk to the man while he eats his breakfast. The fact that neither of them attempt conversation implies that the conflict was serious.