In Bradbury's Fahrenheit 451, the conflicts you're asking about are probably better thought of in terms other than man vs. man. That conflict suggests a grand chess match, or combat, or a fight for survival or victory of some sort.
The conflicts between Millie and Montag and the women and Montag are more conflicts of imagination and thinking and rebirth. The women, including Millie, have no imagination, they do not think, and they are numbly stuck in the mindless status quo of the society.
Montag wants them to use their imaginations, think for themselves, and to be renewed. He wants them, figuratively speaking, to wake up.
These conflicts, again, don't necessarily fit into what is usually thought of as man vs. man.