payload.json

{
  "data":{
    "attributes":{
      "first_name":"TestContact update_custom_fields"
    },
    "relationships": {
      "object-custom-fields": {
        "data": [
          {
           "attributes": {
             "custom_field":"notes",
             "custom_field_value":"awesome"
            } 
          }
        ]
      }
    }
  }
}

response.json

{
  "data": {
    "type": "persons",
    "id": "342",
    "attributes": {
      "prefix": null,
      "first_name": "TestContact update_custom_fields",
      "middle_name": null,
      "last_name": null,
      "preferred_name": null,
      "legal_name": null,
      "suffix": null,
      "date_of_birth": null,
      "contact_types": [],
      "motivations": [],
      "primary_motivation_note": null,
      "is_pre_approved": null,
      "is_primary": null,
      "status": 1,
      "profile_photo": "",
      "stage": null,
      "lead_type": null,
      "is_starred": null,
      "available_to_contact_date": null,
      "buyer_type": null,
      "pre_approved_amount": null,
      "number_in_household": null,
      "background_info": null,
      "touch_count": null,
      "initial_contact_date": null,
      "last_contact_date": null,
      "created_at": "2017-05-26 03:10:19",
      "updated_at": "2017-05-26 03:10:19",
      "company_name": null,
      "company_address": null,
      "company_title": null,
      "company_department": null,
      "primary_email": null,
      "primary_phone": null
    },
    "links": {
      "self": "http://localhost:8000/api/v1/persons/342"
    },
    "relationships": {
      "object-custom-fields": {
        "links": {
          "self": "http://localhost:8000/api/v1/persons/342/relationships/object-custom-fields",
          "related": "http://localhost:8000/api/v1/persons/342/object-custom-fields"
        },
        "data": [
          {
            "type": "object-custom-fields",
            "id": "120"
          }
        ]
      }
    }
  },
  "included": [
    {
      "type": "custom-fields",
      "id": "1",
      "attributes": {
        "name": "notes",
        "custom_field_type": 1,
        "include_in_filters": true,
        "availability": [
          "persons"
        ]
      },
      "links": {
        "self": "http://localhost:8000/api/v1/custom-fields/1"
      }
    },
    {
      "type": "object-custom-fields",
      "id": "120",
      "attributes": {
        "object_id": "342",
        "object_type": "persons",
        "custom_field": "notes",
        "custom_field_value": "awesome"
      },
      "links": {
        "self": "http://localhost:8000/api/v1/object-custom-fields/120"
      },
      "relationships": {
        "custom-field": {
          "links": {
            "self": "http://localhost:8000/api/v1/object-custom-fields/120/relationships/custom-field",
            "related": "http://localhost:8000/api/v1/object-custom-fields/120/custom-field"
          },
          "data": {
            "type": "custom-fields",
            "id": "1"
          }
        }
      }
    }
  ],
  "meta": {
    "constants": {
      "ES": [
        {
          "id": "es_interactions",
          "label": "INTERACTIONS"
        },
        {
          "id": "es_addresses",
          "label": "ADDRESSES"
        },
        {
          "id": "contact_type_ids",
          "label": "CONTACT_TYPE_KEY"
        }
      ]
    }
  }
}

update-payload.json

{
  "data":{
    "attributes":{
      "first_name":"TestContact update_custom_fields"
    },
    "relationships": {
      "object-custom-fields": {
        "data": [
          {
           "attributes": {
             "custom_field":"notes",
             "custom_field_value":"more awesomeness"
            } 
          }
        ]
      }
    }
  }
}