In adults, higher anxiety level related to COVID-19 has been associated with having a pre-existing medical or mental health condition and poor sleep quality. However, no study yet has looked at these links in children. The present study’s main aim was to assess family changes associated with child and parent fears and concerns about COVID-19. We conducted a cross-sectional study among 144 families with children aged 9–12 years during the COVID-19 lockdown period. Families came from Quebec, Canada, and the survey was done in the early stages of the lockdown (April–May 2020). A phone-based survey assessed parent and child COVID-19-related fears and concerns, family-related changes and health issues. Results showed the more fears parents have about COVID-19, the more fears their child also has. Moreover, changes in family sleep habits were associated with parental and child fears and concerns about COVID-19. Reduced access to health services was associated with parental concerns about COVID-19. If another lockdown was to be put in place in the future, it would be important to inform families on the importance of sleep schedules and to maintain or increase health appointments when possible.