In the tmux window where you want turn off the activity alert,
open tmux console (default keys: prefix, :) and
run setw monitor-activity off.
Options of Tmux
There're 3 classes of options: server, session and window. These classes are exclusive: each option belongs to only one of the classes. There is never any inheritance between the option classes.
List server options with show -s.
These options are mostly irrelevant to users. Ignore them.
List default (global) session option values with
show-options -g in tmux console.
Show a specific value with show-options -g base-index.
Set default value of a session option with set -g
(in .tmux.conf, set is abbreviation of set-option).
For example: set -g visual-activity off.
Set this option of a specific session
(which will override the default value) with set.
Unset this option (make it use its default value) with set -u.
List default (global) window option values with show-options -gw.
Show a specific value with show-options -gw monitor-activity.
Show a specific value of a specific window with
show-options -w monitor-activity in this window.
If there's no value being set of this option, there will no output.
Otherwise, you can see a window containing this option and value.
Set default value of a window option with setw -g
(in .tmux.conf, setw is abbreviation of set-window-option).
Set this option of a specific window
(which will override the default value) with setw in this window.
Unset this option (make it use its default value) with setw -u.
Ref:
https://superuser.com/questions/758843/difference-between-global-server-session-and-window-options