Hello and welcome.
Sorry for the delayed response.
Because of the age of the Mac and version of OS X, that version of Safari may have trouble connecting to secure websites (https), because the security certificates contained in Mac OS X Tiger have expired. There isn't any way to update them other than installing a newer version of OS X, which you may be at the maximum version depending on which exact G5 Mac you are using.
When Safari is running, you can click and hold on the Safari icon in the Dock and choose Options > Keep in Dock. Safari will then remain in the Dock when the application is quit.
- What website are you trying to access?
- Does it give the error for all websites?
C