Yes, there have been studies that suggest spending time in nature and engaging in nature-related activities, such as those found in nature activity books, can have positive effects on mental health.
One study published in the journal Scientific Reports found that spending at least 120 minutes (or 2 hours) per week in nature was associated with higher levels of self-reported well-being and lower levels of self-reported anxiety and depression. This was true for people of all ages, genders, income levels, and urban/rural settings.
Another study published in the Journal of Positive Psychology found that engaging in nature-related activities, such as gardening or going for a walk in the park, was associated with greater emotional well-being, vitality, and life satisfaction.
While it's important to note that every child's experience with nature and nature activity books may be different, these studies suggest that incorporating nature into a child's routine may have potential benefits for their mental health.