A

$\because OA=OC$，
$\therefore$点$A$、$C$的坐标为$\left(-c,0\right)$，$\left(0,c\right)$，
$\therefore$把点$A$的坐标代入$y=ax^{2}+bx+c$得，
$ac^{2}-bc+c=0$，
$\therefore c\left(ac-b+1\right)=0$，
$\because c\neq 0$
$\therefore ac-b+1=0$，
$\therefore ac+1=b$.
故选：$A$.