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.