You may activate this creature’s exhaust ability without discarding cards if you just want to put a +1/+1 counter on it.

Exhaust abilities can be activated any time you could normally activate an ability.

If an exhaust ability of a permanent is activated, and then that permanent leaves the battlefield and returns to the battlefield, it becomes a new object so its exhaust ability can be activated again.

If an ability triggers whenever you activate an exhaust ability, that ability resolves before the exhaust ability resolves.