One reason why Connie is complacent is described to the reader in the opening paragraph and refers to Connie's beauty. Note what Connie does in response to taunts of her mother about her thinking she is pretty:
Connie would raise her eyebrows at these familiar old complaints and look right through her mother, into a shadowy vision of herself as she was right at that moment: she knew she was pretty and that was everything. Her mother had been pretty once too, if you could believe those old snapshots in the album, but now her looks were gone and that was why she was always after Connie.
One of the major reasons therefore why Connie is complacent is that she is beautiful, and she knows it, and she takes a kind of perverse smug satisfaction in her beauty and the way that she has what her mother no longer has. From Connie's perspective, being pretty "was everything," and the attitude of her mother in getting at her all the time because of her own loss of beauty only cements the importance of beauty to Connie. Oates therefore presents Connie as being dangerously complacent, which is of course one other aspect to her character that makes her vulnerable.