To generate an Access Token for Twilio Chat, Video, and Voice using Twilio Functions, you need to install dependencies such as Node.js v10, npm or another package manager, and the Twilio Serverless Toolkit plugin. You then initialize a new Twilio Function Service, create a new file called token.js, and gather your credentials including Account SID, Auth Token, and API key. In the token.js file, you add environment variables for these credentials, initialize an Access Token object, generate a grant for the desired API service (Video, Chat, or Voice), and send a response back to the caller with the generated Access Token. Finally, you deploy your function using the Twilio CLI and verify its functionality using curl.