There are several ways to mute a conversation.
1. Hide conversations between users you are following from your timeline
You can hide conversations between accounts you are following from the ︙ icon at the top of the timeline. To use this feature, you must have the timeline added to a tab.
2. Mute mentions to specific users
This is not a feature in the application, but a way to register your username in Keyword Mute and hide it. Note that this will hide all tweets containing that string, not just mentions.
※There is no ability to hide all conversations from the timeline.