I think there are at least a couple of possibilities for what the conflict is in this book. We know for sure that Luke Garner is the protagonist and he is definitely one part of the conflict. But who is he in conflict with?
The first possiblity is that Luke is in conflict with the government. He wants to live and grow and learn, but the government does not want him to exist. In this view, Luke is a representative of people in general and the government stands for society. The conflict represents our struggle as individuals to be what we want to be even if society disapproves.
The second possibility is that there is an internal conflict within Luke. He is struggling to grow up (a coming of age story) and overcome the limitations of the situation he has been born into.
In the book "Among the Hidden" by Margaret Peterson Haddix, the population has become controlled by the government making it illegal for people to have more children than allowed by law. Luke is the families’ third son. He is not supposed to exist. Therefore he must remain hidden away and have no identity of his own. The first conflict is between the law and his parents who violated the law. The second conflict is Luke's need to stay safe from the government. Luke is also in conflict with his parents because he wants to be able to go out and be like everyone else.
Luke eventually meets a girl who is also hidden. He learns about a group of shadow children who are going to stand up against the government. The children are killed but Luke lives. He gets a new identity and later is able to live with different parents as a new person.
I would have to say that the primary conflict over all is man versus the government.