Before you begin, ensure you have a Google Cloud project. Go to the Google Cloud Console, create a new project, and note the project ID. You'll need this project to access Firestore and other Google Cloud services.

Enable the Google PageSpeed Insights API and Firestore API for your project. Navigate to the "APIs & Services" section in the Google Cloud Console, search for both APIs, and enable them. This step allows your application to interact with these services.

To access the PageSpeed Insights API, you'll need an API key. In the Google Cloud Console, go to "APIs & Services" > "Credentials," and create an API key. This key will authenticate your requests to the PageSpeed Insights API.

Write a script to fetch data from the PageSpeed Insights API. You can use a programming language like Python, JavaScript (Node.js), or Java. Here’s a simple Python example:

```python
import requests

API_KEY = 'YOUR_API_KEY'
URL = 'https://www.googleapis.com/pagespeedonline/v5/runPagespeed'
params = {
'url': 'https://example.com',
'key': API_KEY,
}

response = requests.get(URL, params=params)
pagespeed_data = response.json()
```

Replace `'https://example.com'` with the URL you want to analyze and `'YOUR_API_KEY'` with your actual API key.

In the Google Cloud Console, navigate to Firestore. Select "Native mode" and create a database if you haven't already. This database will store your PageSpeed Insights data.

Using the same script, after fetching the data, write it to Firestore. If you're using Python, you'll need the `google-cloud-firestore` library. Install it via pip:

```bash
pip install google-cloud-firestore
```

Here’s how you can write data to Firestore:

```python
from google.cloud import firestore

# Initialize Firestore client
db = firestore.Client()

# Reference to a collection
collection_ref = db.collection('pagespeed_data')

# Add the fetched data to Firestore
collection_ref.add(pagespeed_data)
```

Ensure you have authenticated your environment to access Firestore, typically by setting `GOOGLE_APPLICATION_CREDENTIALS` to point to your service account key file.

Set up a cron job or use a cloud function to periodically run your script and fetch data. For example, on a Unix-based system, you can edit the crontab using `crontab -e` and schedule the script to run at your desired frequency:

```bash
0 * * * * /usr/bin/python /path/to/your/script.py
```

This example runs the script every hour. Adjust the schedule as necessary for your use case.

By following these steps, you can effectively move data from Google PageSpeed Insights to Google Firestore without relying on third-party connectors or integrations.