Yes - to enable this, go to board Settings - Card Template and enable the External ID field.
You can select a few additional options:
for the ID to be visible on fronts of cards or not,
to have the ID editable or not,
for it to be generated automatically for each new task or not,
and for the numbers to be sequential (to start from “1”) or not, in which case the ID will be a random multiple digit number, the same in fact as a task’s internal ID - as seen in an open task’s URL.
Another way to show ID - or any other information from within the card - in closed card view, is to use the Task Summary Power-Up. You’ve got full charge over what your cards are displaying.