To begin, ensure you have the Google Cloud SDK installed on your system. This allows you to access GCS from your command line. You can download it from the [Google Cloud SDK website](https://cloud.google.com/sdk/docs/install). Once installed, authenticate by running `gcloud auth login` and set the appropriate project with `gcloud config set project [YOUR_PROJECT_ID]`.

Determine the bucket and object path from which you want to download data. You can list all objects in a bucket using the command: `gsutil ls gs://[YOUR_BUCKET_NAME]/`. This helps you identify the exact file(s) you need to move.

Use the `gsutil cp` command to download files from GCS to your local machine. For example, run `gsutil cp gs://[YOUR_BUCKET_NAME]/[OBJECT_NAME] [LOCAL_DESTINATION]` to copy a file locally. Ensure the local destination is accurately specified to avoid overwriting existing files.

After downloading, verify the integrity of the data. You can use checksums to ensure the file hasn’t been corrupted during the transfer. Run `gsutil hash -h gs://[YOUR_BUCKET_NAME]/[OBJECT_NAME]` to get the checksum and compare it with the local file using similar tools (like `shasum` on Unix systems).

If the downloaded data isn't already in JSON format, you may need to convert it. Use a script written in a language like Python to read the file and convert its contents to JSON. For instance, if dealing with CSV data, use the `csv` and `json` modules in Python to load the CSV data and then write it to a JSON file.

Once the data is converted to JSON format, write it to a file. Here is a simple Python snippet to do this:
```python
import json

# Assuming `data` is the converted JSON data
with open('[LOCAL_JSON_DESTINATION]', 'w') as json_file:
json.dump(data, json_file, indent=4)
```
Replace `[LOCAL_JSON_DESTINATION]` with your desired file path and name.

Finally, ensure the JSON file is correctly formatted and valid. You can manually inspect the file or use a JSON validator. Python’s `json` module can also be used to read the file back and check for syntax errors. Run a simple script to load the file:
```python
with open('[LOCAL_JSON_DESTINATION]', 'r') as json_file:
try:
data = json.load(json_file)
print("JSON file is valid.")
except json.JSONDecodeError:
print("Invalid JSON file.")
```

Following these steps will allow you to efficiently move data from Google Cloud Storage to a JSON file without relying on third-party connectors or integrations.