Yes, the reading would count toward both if both programs (challenges) are using the same log type.
For example, if Summer Reading awards badges based on minutes, then logging one book would only count toward 1000 Books Before Kindergarten. However, if Summer Reading awards badges based on books, then any book logged during the period of both programs would count toward both.
If you have two concurrent programs that award badges based on different increments, we highly recommend encouraging people to use the Beanstack Tracker app. This mobile app simply tracks reading, and allows readers to enter reading time, book completion, and date for reading sessions.