There are lots of ways to do this, but I'm not sure exactly when you mean by "whenever I am logon".
Basically, if you want the most portable, multilingual way to open a web page with a program you are going to be passing a shell command that looks like this (following your example)
open https://www.google.com
This will open the specified URL (I think it must be a complete URL - including scheme) on the default browser.
Now the "when I am logon" part is what could make things complicated.
If by "when I am logon" you mean whenever you open your browser, then you don't even need to do this you can just reset your browser's homepage.
If you mean whenever you logon to your proxy, you just have to make sure that the first proxy request you send contains the correct request URI. In the case of google the correct URI would be google.com:443. The URI will have the format <hostname>:<portnumber>. The default HTTPS port number is 443. The default HTTP port number is 80.
If you mean whenever you logon to your system then you just need to execute the command open https://www.google.com within a login shell script. Login scripts are system dependent but you may refer to this post if you use a Unix-like or Posix compliant OS. If you use Windows I don't know how to help you but I'm sure there's a way to do it.
If you mean whenever you log into some program you wrote then things get really easy. You just have to make a call to that open command. Almost every language has a built in method for interacting with a shell. You just have to find that method in whatever platform you're using.
I honestly don't know if this will be helpful but good luck!
Edit
How to Run a Batch Shell Script Securely
What you need to do is run the script with a different trustlevel. Since you're using Batch you can do this by executing your script using the runas command (docs). This command allows you to adjust the trustlevel under which the script is executed without changing users. By changing the privileges of the executable, you can prevent it doing anything malicious. All it needs is enough "trust" to execute the open command.
If you need this fully automated throughout your organization, you can wrap your Batch file in a second Batch file that will call the runas command. This wrapper script will not run with reduced privileges but it also won't interact with the internet so you don't have to worry about it as much. You would be able to execute the wrapper script as a login script, and then everything should work.
Honestly I'm not a Windows person but I hope this helps!