We have no way to be sure, but probably not. 

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The way I'm reading the question is, whether the illusion of immortality caused any fear in the Greeks? Well, no source attests to the existence of the [Immortals](https://en.wikipedia.org/wiki/Immortals_(Achaemenid_Empire)) except for [Herodotus](https://en.wikipedia.org/wiki/Herodotus). Opinions are split on his reliability, but it is believed that Herodotus did not personally speak Persian.

> In Paligaro's view, Herodotus or his informant confused here the word anusya ("follower", "adherent") with the word "anausa ("immortal").
>
>  <sub>Dandamaev, Muhammad A., and Vladimir G. Lukonin. *The Culture and Social Institutions of Ancient Iran.* CambrSidge University Press, 2004.</sub>

We have no reason to disbelieve that the Persian king was guarded by a unit of companions. Macedon had the [Companion Cavalry](https://en.wikipedia.org/wiki/Companion_cavalry), for instance. It also stands to reason that they were an elite force. However, maintaining a unit at constant strength is hardly a frightening feat in and of itself. Persian propaganda *could* have hyped it up as "immortality"; but if that were the case, we'd expect it to be reflected in the unit's name, or nicknames.

That the only source to call the unit Immortals may well have been an error, suggests they didn't. By extension, therefore, the Greeks wouldn't have known to fear its "technically immortal" status.

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More generally speaking, the account given by Herodotus hardly suggests fear for the Immortals in Greece, whether for their combat ability or the "immortality". There is certainly no trace of it in the paragraph cited in the question. In fact, Herodotus seemed much more concerned with the Immortals' *appearances* than either their reputation or fighting prowess.

From [*The Histories*](https://en.wikipedia.org/wiki/Histories_(Herodotus)), 7.83.3,

> The Persians showed the richest adornment of all, and they were the best men in the army. Their equipment was such as I have said; beyond this they stood out by the abundance of gold that they had. They also brought carriages bearing concubines and many well-equipped servants; camels and beasts of burden carried food for them, apart from the rest of the army.

The idea that they were the best of the Persian army was mentioned only in passing, and the greater part of the paragraph was spent describing their splendour and status.

Moreover, Herodotus goes on to list the Persian cavalry and nomadic units in the Persian army. The over impression is that he was describing an [order of battle], not putting the Immortals on a pedestal as a unit to be feared.

Since *The Histories* is our only source, all the evidence we have suggests the Greeks did not particularly fear the Immortals at all.