To begin, you need to have access to Intercom's API. Log in to your Intercom account, navigate to the 'Developers' section, and create a new API key. Ensure that it has the necessary permissions to access the data you want to export.

Ensure you have the necessary tools installed. You'll need a command-line tool like `curl` to make HTTP requests and a programming environment (such as Python or Node.js) to process the data. If using Python, ensure `requests` library is installed by running `pip install requests`.

Use `curl` or a programming script to make a GET request to the Intercom API endpoint relevant to the data you need (e.g., users, conversations, etc.). For example, with `curl`, you might use:
```bash
curl https://api.intercom.io/users -H "Authorization:Bearer "
```
Replace `` with your actual API token.

The data received from Intercom will be in JSON format. Use your chosen programming language to parse this JSON data. For Python, you can use the `json` module:
```python
import json
data = json.loads(response.text)
```

Depending on your requirements, you may need to filter or restructure the data. For example, you might want to extract only certain fields from each user object. This can be done programmatically:
```python
filtered_data = [{"id": user["id"], "email": user["email"]} for user in data["users"]]
```

Once your data is filtered and structured as needed, save it to a local JSON file. In Python, this can be done using:
```python
with open('intercom_data.json', 'w') as file:
json.dump(filtered_data, file, indent=4)
```

After saving the data, verify the JSON file to ensure it contains the correct information. Open the file and check the contents. Also, ensure that the data is stored securely, especially if it contains sensitive information. Consider setting appropriate file permissions and encrypting the file if necessary.

By following these steps, you can successfully export data from Intercom to a local JSON file without relying on third-party connectors or integrations.