If you'd like to turn off notifications and alerts for a particular Facebook conversation, you can do it right from the chat settings. Here's how to stop chat tabs from showing up on your screen.
Sign in to Facebook
and select the conversation you'd like to mute. Click on the cog
icon displayed at the upper right corner of the chat window and select Mute Conversation
A dialogue window titled Mute Conversation
will open. Choose how long you'd like to keep notifications muted and click on Mute
to apply your new settings:
Messages from the muted contacts or groups will be forwarded to your inbox.