Begin by enabling API access in your Zendesk account. Go to the Admin Center, select "Channels" and then "API". Ensure that API access is enabled and note down your subdomain, email, and API token. These credentials will be used to authenticate requests to the Zendesk API.

Access your Google Cloud Console and create a new project if you haven't already. Enable Firestore by navigating to Firestore in the left-hand menu and selecting "Create Database". Choose a start mode (production or test) and note the database settings. Ensure you have the necessary permissions to write data to Firestore.

On your local machine or server, ensure you have a development environment set up with Node.js or Python (or any preferred programming language that supports HTTP requests and JSON). Install the required libraries for making HTTP requests and for interacting with Firestore. For example, in Node.js, you might use Axios for HTTP requests and the Firebase Admin SDK for Firestore.

Write a script to fetch data from Zendesk using its REST API. Use HTTP GET requests to access the desired endpoints (e.g., tickets, users). Authenticate your requests using Basic Auth with your Zendesk email/token. Parse the JSON response to extract the data you need.

Example in Node.js:
```javascript
const axios = require('axios');
const zendeskDomain = 'your_subdomain.zendesk.com';
const zendeskEmail = 'your_email/token';
const zendeskToken = 'your_api_token';

async function fetchZendeskData() {
const response = await axios.get(`https://${zendeskDomain}/api/v2/tickets.json`, {
auth: {
username: zendeskEmail,
password: zendeskToken
}
});
return response.data.tickets;
}
```

Process the fetched data to fit the structure required by Firestore. This may involve transforming the data format or filtering out unnecessary fields. Ensure that your data is in JSON format, as this is compatible with Firestore.

Use Firebase Admin SDK to authenticate and initialize Firestore in your script. Download the service account key from your Google Cloud project and load it in your script for authentication.

Example in Node.js:
```javascript
const admin = require('firebase-admin');
const serviceAccount = require('./path/to/serviceAccountKey.json');

admin.initializeApp({
credential: admin.credential.cert(serviceAccount)
});

const db = admin.firestore();
```

Write the processed data to Firestore by specifying the collection and documents you want to create or update. Use Firestore methods to add data, ensuring that data types are compatible with Firestore's data model.

Example in Node.js:
```javascript
async function uploadToFirestore(tickets) {
const batch = db.batch();
tickets.forEach((ticket) => {
const ticketRef = db.collection('tickets').doc(ticket.id.toString());
batch.set(ticketRef, ticket);
});
await batch.commit();
console.log('Data successfully written to Firestore');
}

async function main() {
const tickets = await fetchZendeskData();
await uploadToFirestore(tickets);
}

main().catch(console.error);
```

By following these steps, you can transfer data from Zendesk Support to Google Firestore without relying on third-party connectors or integrations, using the powerful capabilities of APIs and custom scripting.