Could a Gelatinous Cube attune to a headband of intellect?

Not with a headband

Nothing prohibits a monster, or 'NPC', from using magical items, and those meant to be worn do adapt to their users, somewhat, changing size for example. But a fundamental assumption is that magic items that go on body parts can be used only by creatures with the appropriate parts:

The DMG (p. 140) says in Wearing and Wielding Items:

A magic item meant to be worn must be donned in the intended fashion: boots go on the feet, gloves on the hands, hats and helmets on the head, and rings on the finger...
When a nonhumanoid tries to wear an item, use your discretion as to whether the item functions as intended. A ring placed on a tentacle might work, but a creature with a snakelike tail instead of legs can't wear boots.

As a creature without a head, a gelatinous cube cannot use a headband, per se. You ask whether the gelatinous cube can attune to the headband 'or similar item'. If 'similar item' means 'an item that also raises intelligence but doesn't necessarily go specifically on the head', then perhaps.

There is some leeway here for varying body types - if "A ring placed on a tentacle might work", then the cube might not have to have exactly a head as long as it had a body part that was 'head-like'.

The Sage Advice Compendium explains:

Is the grappling rule in the Player’s Handbook usable by a handless creature? The grappling rule (PH, 195) was written for a grappler with at least one hand, but a DM can easily adapt the rule for a handless creature that has a bite or an appendage, such as a tentacle, that could reasonably seize someone. A wolf, for example, could plausibly try to seize a person with its bite, and the animal wouldn’t be able to use its bite attack as long as it held onto the person.

Thus a true head should not be required to attune to the headband, so long as a willing DM rules that some part of the cube is reasonably head-like. Alas, the cube is...a cube:

Ooze Cube. The cube takes up its entire space. Other creatures can enter the space, but a creature that does so is subjected to the cube's Engulf...

(and see the official illustration)

The cube has no parts that could plausibly be considered a head. At best, a lenient DM might permit it to wear a gauntlet or a boot on its pseudopod, provided the magic item was immune to acid damage.

In case you are considering it, polymorph won't likely work, either. It should be easy enough to transform the cube into a beast that does have a head (and most of them do), and once there have it immediately start attuning, a process which will be complete just as the polymorph ends. Unfortunately for your plan, when the now-attuned Int 19 beast returns to its natural cube form, it will immediately break attunement, as the attunement rules say

A creature's attunement to an item ends if the creature no longer satisfies the prerequisites for attunement...

and it will return to being a normal cube, without a head, and thus no longer satisfy the prerequisite for attuning to the headband.

Now, Jack suggests that "If you think your game would be enhanced by the cube wearing a headband...Just go ahead and perch that headband up there." If you and your DM are open to a Rules as Fun approach (RAF), then sure. As a slightly-less-absurd step than 'throw the headband on the gelatinous cube and it becomes intelligent', you might also consider 'polymorph it first but then allow the attuned headband to continue to work when it returns to its natural form'. If I included such a creature in my game, I would strongly consider removing the cube's inherent immunity to the charmed and frightened conditions once it had attained sentience.

Kirt's very good answer is good as far as it goes. It's worth considering specifically from the player's and the DM's point of view, though.

If you're a player

Rules-as-written, as Kirt notes, no head, no way to wear a headband.

If for some reason you really want to pull this off, you'll need to work with your DM. Talk to them out of session, maybe, just to give them a chance to consider it.

If you're the DM

If you think your game would be enhanced by the cube wearing a headband, don't feel constrained by the rules at all. D&D is full of absurd things. Published adventures are full of absurd things. You should not feel constrained at all at perching a headband of intellect on top of a gelatinous cube, if that serves the game. Just go ahead and perch that headband up there.

There could be a lot of fun here. As SeriousBri noted in a comment, once the Gelatinous Cube gets a headband:

another thing to consider would be if an intelligence 19 creature (likely one of the smartest in the party) actually would want to hang around with a group of danger-seeking lunatics who thought of it as a pet or follower.

So, does Algernon want to hang out with the party? Maybe Algernon becomes the brains of the party. Or maybe Algernon ditches them entirely. An intelligence of 19 is pretty smart. Maybe Algernon strikes out on its own. Maybe Algernon adds a Hat of Disguise and becomes some village's reclusive sage . . . .

Is the creature smart enough?

A Gelatinous Cube has an Intelligence of 1, which is equal to or lower than all the beasts in the Monster Manual, like the Allosaurus (INT 2), Cat (INT 3), Giant Octopus (INT 4), Frog (INT 1), or Raven (INT 3). Only six beasts have more than 5: Ape (INT 6), Giant Ape (INT 8), Giant Eagle (INT 8), Giant Elk (INT 7), Giant Owl (INT 8), Giant Vulture (INT 6).

Attuning requires you to stay focused on the Magic Item, keep in contact with it, and, as shown in the example for a wondrous item, meditate on it (DMG, p. 136, emphasis mine):

Attuning to an item requires a creature to spend a short rest focused on only that item while being in physical contact with it (this can't be the same short rest used to learn the item's properties). This focus can take the form of weapon practice (for a weapon), meditation (for a wondrous item), or some other appropriate activity. If the short rest is interrupted, the attunement attempt fails. Otherwise, at the end of the short rest, the creature gains an intuitive understanding of how to activate any magical properties of the item, including any necessary command words.

Intelligence 1 means, at least to me, a creature with no ability to use logic or reasoning, and with no knowledge to understand what to do with a Headband of Intelligence or other Magic Items in order to attune to them.

DM's ruling

Of course a DM might decide otherwise, but this answer is just meant to give a different point of view based on how attunement works, in addition to the good answers and reasons already given.