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I'm looking for an English word that describes something comfortable: a safe space that one prefers to chill out in quiet solitude, and something that depicts a sentiment of familiarity.

I would like for the word to be informal, as per something you would say to a good friend of yours: 'And this is my little corner here.' 'Little corner' can be a good candidate but I'm looking for a single-word term.

By reading up on the thesaurus and the dictionary, I stumbled upon the word 'nook'.

So, I wonder, is the use of this word by itself appropriate? Or is it unidiomatic? I always used to hear it in the saying 'nook and cranny', and I wonder whether it's uncommon to have only 'nook' in a sentence.

Yes, you can use nook without the cranny, for nook merely means "a corner or recess, especially one offering seclusion or security." (Oxford Dictionary) See here for sentence examples of nook by itself. As to whether it is common, that is something I wouldn't concern myself with. Commonality, or lack thereof, should not dictate whether something is used or not. Usage of language should be dictated only by communicative qualities; that is, whether the term or word in question adequately communicates an idea, regardless of how old or common it is. This is just my opinion, however.