To use Putty with Google Cloud Compute Engine VM, we need first to install the Google Cloud SDK. You can refer the following article for a step by step guide to install Google Cloud SDK on Windows.
Step 1: First, we need to login in with Google Cloud Gmail account in the default browser.
Step 2 Open Command Prompt and execute the following command
gcloud auth login
Above step will open your default browser. Ensure you selected the correct google cloud account.