Add it after the /write command.

Code:
on *:text:!addquote*:*: {
  if ($2 == $null) { .notice $nick You need to include a quote to add. }
  if (%quote.delay) {
    .msg $nick Please try again after random delay of: $duration(%quote.delay)
    return 
  }
  else {
    var %delay = $rand(25,35)
    set -u [ $+ [ %delay ] ] %quote.delay %delay
  }
  write quotes.txt $2-
  ;ADD THIS NEW LINE HERE
  ;.NOTICE $nick added a new quote - done.
}


And your on input.

Code:
elseif ($1 == !addquote) {
    if ($2 == $null) { echo -a You need to include a quote to add. }
    else write quotes.txt $2-
    ;ADD THIS NEW LINE
    .notice $me Done.
  }


Edit: And Riamus beat me to it, but he meant .msg $me instead of .msg $nick for the on text. But then, you personally suggested noticing yourself, so replace that with .notice.