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:

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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.