You can archive an entire conversation or individual messages with a contact to remove it from your Conversations view.

  1. Tap Conversations in the lower toolbar. 
  2. Tap the desired conversation to open it. 
  3. Tap the Options icon (three vertial dots) in the upper toolbar. 
    • Select Archive Conversation to remove the entire timeline from you conversations view. 
    • Select Edit Conversation to remove specific messages, then select the desired items and tap the Delete icon in the upper toolbar.

Keep in mind that the next call, text, or email sent by either party will cause their timeline to reappear in your Conversations view. Also see How do I restore archived conversations? (iOS only). If needed, it is also possible to block calls from a contact.