Yes, but they do not add to the Rod’s energy.
The Rod of Absorption says:
While holding this rod, you can use your reaction to absorb a spell that is targeting only you and not with an area of effect.
We observe in the rules for spellcasting that cantrips are spells:
A cantrip is a spell... A cantrip's spell level is 0.
Cantrips are spells, so they are valid for use with the Rod of Absorption, but add 0 levels of energy to the Rod each time it absorbs a cantrip, since a cantrip’s spell level is 0.
It may not be feasible to use your Rod only for cantrips.
It sounds like a good idea at first, using the Rod only to absorb cantrips. But this probably isn't entirely feasible. If your DM just announces what spells the NPCs are casting, and is cool with you just cancelling them with your Rod, then it should work. But if you're going to try to do this at my table, I'm going to stop saying, "The enemy wizard is going to cast firebolt at you", and will rather describe the spellcasting. Then, if you want to try to use the Rod for cantrips, I will use the optional rules from Xanathar's Guide to Everything for Perceiving a Caster at Work:
Sometimes a character wants to identify a spell that someone else is casting or that was already cast. To do so, a character can use their reaction to identify a spell as it’s being cast, or they can use an action on their turn to identify a spell by its effect after it is cast.
If the character perceived the casting, the spell’s effect, or both, the character can make an Intelligence (Arcana) check with the reaction or action. The DC equals 15 + the spell’s level. If the spell is cast as a class spell and the character is a member of that class, the check is made with advantage. For example, if the spellcaster casts a spell as a cleric, another cleric has advantage on the check to identify the spell. Some spells aren’t associated with any class when they’re cast, such as when a monster uses its Innate Spellcasting trait.
This Intelligence (Arcana) check represents the fact that identifying a spell requires a quick mind and familiarity with the theory and practice of casting. This is true even for a character whose spellcasting ability is Wisdom or Charisma. Being able to cast spells doesn’t by itself make you adept at deducing exactly what others are doing when they cast their spells.
So using your Rod only for cantrips would probably not be feasible at my table. It would at least cost your reaction and wouldn't succeed all the time.
After the Rod is full, it becomes the Rod of Cantrip Busting
Once the Rod is full, there is nothing to prevent it from still absorbing cantrips. The description says:
The rod can absorb and store up to 50 levels of energy over the course of its existence. Once the rod absorbs 50 levels of energy, it can't absorb more.
It says you cannot store more energy, not that it can no longer absorb spells. Since a level 0 spell stores 0 levels of energy, it probably still works.
There is an alternate interpretation that invalidates all of this.
The description does say:
If you are targeted by a spell that the rod can't store, the rod has no effect on that spell.
One could arguably understand this to mean that a 0 level spell is a spell the rod cannot store. Take this way, cantrips would actually never work with the Rod of Absorption.