Short answer: Yes, you can if you have the valid accessToken. You can call the Extension Info endpoint using the accessToken.
It would be much easier to use one of our official SDKs to do this. Here is some code snippet using the Node JS SDK.
var rcsdk = new RC({
    server: 
https://platform.devtest.ringcentral.com,
    appKey: your_app_key,
    appSecret: your_app_secret
});
var platform = rcsdk.platform();
var data = platform.auth().data();
        data.token_type = "bearer"
        data.expires_in = 86400
        data.refresh_token_expires_in = 86400        
data.access_token = "THE accessToken"        platform.auth().setData(data)
        platform.get('/account/~/extension/~/')
          .then(function(response) {
              var jsonObj =response.json();
              var extensionNumber = jsonObj.extensionNumber              var userName = jsonObj.name          })
          .catch(function(e) {
              console.log("Failed")
              console.error(e);
              throw e;
          });
Hope this helps!
Phong Vu
                
    
            Here is the screenshot of the API Explorer.

You just click the Try It button without providing the extension Id.
If it does not work with you code, what error did you get? Can you post the response from the server?