"Quiet alertness" means being attentive and aware without making a lot of noise or being overly active. It implies being calm and focused while still being mindful of one's surroundings. Full definition
The behavioral state of quiet alertness also gives parents a better opportunity to interact with their baby. (thefashionablebambino.com)
I felt confident about two things: breastfeeding, and putting him in the wrap we'd bought where he almost inevitably fell asleep, or kept to such a state of quiet alertness that he might as well have. (romper.com)
They are learning more because babies spend more time in a quiet alertness and they are able to connect with the new world around them when they are not crying. (ridgemeadowsdoulaservices.com)