class Family(models.Model):
    name = models.CharField(max_length=100)

class Category(models.Model):
    name = models.CharField(max_length=100)

class Trend(models.Model):
    name = models.CharField(max_length=100)
    area = models.ForeignKey(Area, related_name='subcategories')
    families = models.ManyToManyField(Family, related_name='subcategories')

SELECT
  cat.*,
  subcat.*,
  CASE
  WHEN tr.id IN (
    SELECT tr.id
    FROM family fam
      INNER JOIN subcategoryfamilyrel rel ON fam.id = rel.family_id
      INNER JOIN subcategory subcat1 ON rel.trend_id = subcat1.id
    WHERE
      fam.id = {family.id}
  ) THEN 1
  ELSE 0
  END AS active
FROM subcategory subcat
  INNER JOIN category cat ON subcat.category_id = cat.id;

Subcategory.objects.annotate(
    active=Case(
        When(id__in=profile.family.subcategories.all().values('id'), then=True),
        default=Value(False),
        output_field=BooleanField()
    )
)

[
  {
    "id": 1,
    "name": "Русский язык",
    "active": true,
    "area": {
      "id": 1,
      "name": "Школа",
      "color": "123456"
    }
  },
  {
    "id": 2,
    "name": "Математика",
    "active": ,
    "area": {
      "id": 1,
      "name": "Школа",
      "color": "123456"
    }
  },
  {
    "id": 6,
    "name": "Уборка",
    "active": true,
    "area": {
      "id": 2,
      "name": "Дом",
      "color": "ABCDEF"
    }
  }
]

[
  {
    "id": 1,
    "name": "Школа",
    "color": "123456",
    "subcategories": [
      {
        "id": 1,
        "name": "Русский язык",
        "active": true
      },
      {
        "id": 2,
        "name": "Математика",
        "active": false
      }
    ]
  },
  {
    "id": 2,
    "name": "Дом",
    "color": "ABCDEF",
    "subcategories": [
      {
        "id": 3,
        "name": "Уборка",
        "active": true
      }
    ]
  }
]