default.json.twig

{% set payload = {frontmatter: page.header, content: page.content}  %}
{{ payload|json_encode|raw }}

home.json.twig

{% set items = {} %}
{% for child in page.collection() %}
    {% set child_output %}
        {% include 'default.json.twig' with {page: child} %}
    {% endset %}
    {% set items = items|merge({ (loop.index0): child_output}) %}
{% endfor %}

{% set payload = {
    name: page.header.title,
    route: page.route,
    type: page.template,
    items: items,
    content: {
        body: page.content
    }
} %}

{{ payload|json_encode|raw }}

output.json

{
  "name": "Home",
  "route": "/",
  "type": "home",
  "items": [
    "        {\"frontmatter\":{\"title\":\"Team\"},\"content\":\"<p>Content<\\/p>\"}    "
  ],
  "content": {
    "body": "<p>Content stuff here</p>"
  }
}