Much of early literacy education happens at home with caretakers reading books to young children. Prior research demonstrates how having dialogue with children during co-reading can develop critical reading readiness skills, but most adult readers are unsure if and how to lead effective conversations. We present ContextQ, a tablet-based reading application to unobtrusively present auto-generated dialogic questions to caretakers to support this dialogic reading practice. An ablation study demonstrates how our method of encoding educator expertise into the question generation pipeline can produce high-quality output; and through a user study with 12 parent-child dyads (child age: 4-6), we demonstrate that this system can serve as a guide for parents in leading contextually meaningful dialogue, leading to significantly more conversational turns from both the parent and the child and deeper conversations with connections to the child's everyday life.