The exact mechanism depends on what shell is running in the "terminal session". For the BASH shell, the man page for "bash" says:
MAILCHECK
Specifies how often (in seconds) bash checks for mail. The
default is 60 seconds. When it is time to check for mail, the
shell does so before displaying the primary prompt. If this
variable is unset, or set to a value that is not a number
greater than or equal to zero, the shell disables mail checking.
so setting `MAILCHECK=-1` in your `.bashrc` file would do it. Other shells have man pages with similar advice. (My `bash 5.0.17` refuses to let me set the variable to a non-integer unless I first `unset` it, so the man page is incomplete about using "not a number".)