Begin by obtaining API access to Google PageSpeed Insights. Go to the Google Cloud Console, create a new project, and enable the PageSpeed Insights API. Generate an API key that will be used to authenticate your requests to the PageSpeed Insights API.

Use a programming language like Python to make HTTP requests to the PageSpeed Insights API. You can use the `requests` library to send a GET request to the API endpoint, appending your API key and desired parameters (such as the URL you want to analyze) to the request URL. Parse the JSON response to extract the performance data you need.

Once you have the data, transform it into a format that is suitable for storage in Convex. This might involve restructuring JSON objects, filtering out unnecessary information, and ensuring that the data types match what is expected by Convex. You will likely convert the data into a format like JSON, CSV, or another structure that can be easily ingested by Convex.

If you don’t have one already, set up a Convex environment. This involves creating an account on the Convex platform, setting up a new project, and configuring any necessary settings. Familiarize yourself with Convex's data ingestion methods to understand how data can be uploaded or transferred into the system.

Develop a script that will automate the transfer of data to Convex. This script should take the transformed data from Step 3 and use Convex's API or upload functionality to insert the data into the desired location in your Convex project. Ensure that you handle authentication and any required headers or tokens as specified by Convex's documentation.

Before automating the process, perform a series of tests to ensure that data is being transferred correctly from Google PageSpeed Insights to Convex. Verify that the data is accurate, complete, and in the correct format once it reaches Convex. Adjust your script as necessary to handle any errors or edge cases.

Once you have confirmed that the data transfer process works as expected, set up a cron job or use a task scheduler to automate the script execution at regular intervals. This will ensure that your Convex environment is continually updated with the latest data from Google PageSpeed Insights without manual intervention.