Consider the following situation (from my perspective as the DM):

• One of my PCs casts a spell on an NPC (or monster etc.) which forces the target to make, say, a Wisdom save, or else be frightened.




• Before the NPC rolls for his save, the divination wizard PC announces that she will use a low Portent die to give him a very low score on the save.




• Unknown to either the PCs or the players, this NPC has a magical item that prevents him from being frightened, so regardless of the roll of the die, he will be unaffected by the spell.




• If I announce that the NPC is not frightened, the players (and PCs) will know that the NPC either has a very high Wisdom modifier, or has some item or ability that prevents him from being frightened.




• I would rather the players and PCs did not know this. Also, the NPC would rather the PCs did not know this.

Can the NPC tell that he has been "Portented"?

If so, he could choose to pretend that he had failed the save i.e. pretend to be frightened. Depending on the circumstances, this might be preferable for the NPC to giving away the existence of his magical item. It might also be preferable for me as the DM, given that I want to keep this item a secret...

No, there is no way to know that Portent was "used"

Portent is an unusual feature in that it makes an implied distinction between the player and their character. Obviously, it is the player's choice when to replace a roll with a Portent die, but within the fiction, this does not correspond to any action or even any thought or intent on the character's part. All the character does is witness the fact that the glimpse of the future they received earlier that day has indeed come to pass.

This interpretation is supported by the first sentence of the feature, which is the only part that describes how it works rather than the mechanics of using it:

Starting at 2nd level when you choose this school, glimpses of the future begin to press in on your awareness.

Also note that no action or reaction of any kind is required to use the feature. If Portent represented an active manipulation of probability by the diviner, it would make more sense for it to require a reaction to do so.

Fear immunity is not so unusual

To address your specific problem, immunity to fear is not a terribly rare effect, so that alone isn't necessarily going to tip off your players that a special magic item is in play. Furthermore, if your players do infer that a magic item is protecting NPC from fear, they will most likely assume that the item does that and nothing more. From your question, it seems like the item in question probably does a good deal more than simply protect from fear. But if you're really worried about your players figuring it out, I think the best advice is to make sure you can still tell an interesting story if they do so, even if it's different from the story you had planned.

No.

The Portent feature doesn't have any visible or otherwise detectable in-world component. It's debatable whether even the Diviner knows about the Portent.

However, your NPC might be able to tell which spell is being cast.

An optional rule in Xanathar's Guide to Everything allows a creature to use a reaction to make an Arcana check to know which spell the PC is casting (DC is equal to 15 plus the spell's level). You could make this roll behind the screen to allow the NPC to know that the spell would normally cause fear, an effect they're immune to, and then act as if they're frightened.

Probably not.

To understand the workings of the feature, here is Diviner's Portent from p.116 of the Player's Handbook:

Starting at 2nd level when you choose this school, glimpses of the future begin to press in on your awareness. When you finish a long rest, roll two d20s and record the numbers rolled. You can replace any attack roll, saving throw, or ability check made by you or a creature that you can see with one of these foretelling rolls. You must choose to do so before the roll, and you can replace a roll in this way only once per turn
Each foretelling roll can be used only once. When you finish a long rest, you lose any unused foretelling rolls.

As seen from here, the Divination Wizard doesn't actually alter the future but instead sees glimpses of it. Since all rolls of a die are independent of each other and equally likely to land on any of the numbers 1 through 20, the act of you replacing a creature's roll with a Portent die isn't equivalent to, say, imposing disadvantage on a roll, since what has essentially happened is that the die roll for that particular moment was already rolled when the Divination Wizard took a long rest.

It's ultimately up to how you roleplay it as to how powerful this can become, however. An intelligent creature, like a goblin, might notice that a certain wizard is of the divination school by the nature of their spells, and would realise that diviners can see glimpses into the future. As a result, it may logically pretend to fail saving throws against conditions despite immunity to them to throw off the players.