What is the conflict in the beginning of The Egypt Game?

I think you are refering
to the second chapter of this excellent book, because the book actually starts in
media res, or in the middle of things, and then cuts to the beginning in the second
chapter. This chapter introduces a very important character and her central conflict. April Hall
has been sent to live with her grandmother by her mother, however April deeply resents this and
feels that this is just a temporary arrangement, and that her mother will send for her very
soon. Note how this second chapter introduces the scenario:

Exactly one month before the Egypt Game began in the Professor's backyard she had come,
very reluctantly, to live in teh shabby splendour of an old California-Spanish apartment house
called the Casa Rosada. She came because she had been sent away by Dorothea, her beautiful and
glamorous mother, to live with a grandmother she hardly knew, and who wore her grey hair in a
bun on the back of her head.

April's conflict thus lies
in the way that she has been effectively sent away by her mother to live with a stranger. She
refuses to settle down in this place, as this would signal that she recognises her mother is
highly unlikely to send for her and ask her to come and live with her
again.