Adding a button near profile

The post meta output, which can include the post author, is theme dependent. There may be a filter you can use or a pluggble function to redefine. You should ask your theme’s devs for specifics.

That will give you a link button, but chat functionality does not occur by default, you would need some sort of plugin that offers that functionality. WP does have comment functionality for posts, which is similar but different than chat.