I think that Laura's exchange with her mother in Scene two helps to reveal much within her that is in conflict with the outside world. In this confrontation, there is much gained about Laura and how she sees the world:
I went in the art museum and the bird-houses at the Zoo. I visited the penguins every day! Sometimes I did without lunch and went to the movies. Lately I’ve been spending most of my afternoons in the Jewel-box, that big glass house where they raise the tropical flowers.
Laura reveals herself to be fundamentally different from social expectations. It is here where Laura's conflict between the identity within her and the identity that she is meant to embrace is evident. Laura lies in conflict between both realities. She finds happiness in a path that is not socially sanctioned. This causes conflict because of its fundamentally divergent nature. It also causes conflict because it lies in opposition to her mother's wishes.
Tom's character is fraught with conflict. In Scene six, Tom reveals the conflict that will end up galvanizing him into action at the end of the drama:
I’m starting to boil inside. I know I seem dreamy, but inside—well, I’m boiling! Whenever I pick up a shoe, I shudder a little thinking how short life is and what I am doing!—Whatever that means, I know it doesn’t mean shoes—except as something to wear on a traveler’s feet
In this quote, Tom shows himself to be ready to move on. The life that he has led is slowly reaching its conclusion and something new is needed. Tom's conflict is that he fundamentally detests the life he leads and feels that he can no longer continue in such a capacity. It is here where conflict is evident. His "starting to boil inside" helps to reveal the internal conflict that Tom feels between what he is supposed to do and what it is that he really wants to do.