Even though there is no option to edit chat window messages by default, you are able to customize them using Custom CSS. To do so, navigate to Configuration > Chat > Chat buttons > Edit your chat button. Then in the "Chat window" tab scroll down to the "Custom CSS" section and add the following CSS code:
.MessageNotification.MessageNotification-Unread {
background: green !important;
color: white !important;
}
Do not forget to APPLY the CSS code and SAVE the chat window.