I am unaware of the feature for bookmarking a chat on Stack Overflow. I saw some users having a bookmark of a particular part of chat in the room. So, from where can we bookmark a particular part of chat?
It's explained in the Chat FAQ, in the Can I share a link to a specific message? Or range of messages? section:
Sure; if you click the menu button against any message, you get the "permalink" option:
This gives you a link directly to that message, highlighting that message for clarity. If you want to create a link to a group of messages, you can bookmark a block of conversation by choosing "create new bookmark" from the room menu:
Then just follow the simple on-screen instructions to select the messages you want to include. You need at least 100 reputation to create a new bookmark.
Hover over a chat message and click the small arrow that appears.
There will be a link at the top of the pop-up that appears. Use your browser to bookmark this link.