The external conflict in Anne of Green Gables is Anne's struggle to be accepted by the traditional and small rural town of Avonlea.
When Anne first appears at the train station, Matthew is flabergasted because she was supposed to be a boy: Matthew and Marilla had requested a boy to help work on the farm to aid Matthew as his health deteriorated. While Matthew comes to accept Anne fairly quickly for her funny way of talking, Marilla is more practical.
Marilla rejects Anne until she comes to feel empathy for her situation. If Anne can't live with the Cuthberts, she will be forced to become a working girl for a woman who has too many children and who abuses the girls who are sent to help her with the care of her children and other household duties. Anne has worked in similar situations in the past, and, in the end, Marilla cannot bear to send her (or Anne's hopes of having a family and being cared for for the first time in her life) away.
Marilla also takes a firm interest in Anne's spiritual well-being, regarding her "heathen" demonstrations as God's charge to Marilla to help shape Anne's character, as she recognizes no one has even taught Anne to pray (much less to live with virtue).
Rachel Lynde, Marilla's best friend and nosy neighbor, also rejects Anne for being an orphan as do many of the girls at school and the people at church. There is a stigma surrounding orphans that Anne must overcome to find acceptance.
Heartbreakingly, Diana Barry's mother is another character who rejects Anne. She considers her as a bad influence on Diana, who is the one character who loves Anne from the start besides Matthew. However, when Anne rescues Mrs. Barry's youngest child, she wins back her approval. This is just one example of how Anne must "earn" the approval of the judgemental and close-minded people in this novel.
Her internal struggle aligns closely with this as she struggles with her temper, being unaccustomed to social etiquette, and being too "romantic" in her notions to satisfy the cultural expectations of the small town. Her temper and inclination to act dramatically when wounded is gradually overcome by the exerted efforts of Marilla and Anne's eventual understanding that the Cuthberts care for her: she eventually realizes that they are only teaching her their customs in order to help her be her best self.
In learning to walk with character and by serving others in extraordinary ways when challenges arise for them, Anne overcomes her internal and external obstacles.