The Doylist Explanation: No
There was never any one fixed version of Quenya or Sindarin that Tolkien definitively 'finished'. His work on the languages was a lifelong endeavour, with the languages evolving even to the point of his death.
The one bit of the Legendarium that we can be fairly sure was originally written in Quenya, i.e. the Namárië, also called Galadriel's Lament - was revised multiple times when it came to the version of language and grammar being used.
So, given that he never really 'finished' the languages, which were his first passion and hobby (with the phenomenally successful books being a secondary project), it would have been difficult for him to write any significant parts in the languages.
Namarie, in fact, is known to be one of the longest continuous texts in Quenya that was ever written by Tolkien. So it seems unlikely that any significant parts of the legendarium could have been conceived entirely in an Elvish tongue.
In a way, however, since the Legendarium exists primarily to give 'flavor' and 'history' to the languages, which were Tolkien's first love, we can also very easily assume that they were always in the forefront of his mind when writing, even if written in English. He certainly thought of the names of things and people that he was 'translating' from. In Letter #190 he becomes quite angry at the suggested Dutch translations of nomenclature for a Dutch version of the books. And then there's the following well-known bit of info:
The real 'historical' plural of dwarf (like teeth of tooth) is dwarrows, anyway: rather a nice word, but a bit too archaic. Still I rather wish I had used the word dwarrow.
- Letter #17 regarding dwarves vs dwarfs
Which leads up to...
The Watsonian Explanation: Yes, Every Bit of it.
As you mention, the conceit of Tolkien's Legendarium is that Tolkien is merely 'translating' all of the works from various languages (mainly Westron and Quenya) into English. But many of the posthumously published writings make it clear that it wasn't a conceit imposed on the work once it was finished, instead much of the work was written with the device that it was merely a translation as a intrinsic part of the nature of the work.
This is used to explain in-universe things like: the two versions of Bilbo winning the Ring; the huge difference in styles of writings between the Hobbit (aka Bilbo's diary); the LotR (written as a semi-diary, semi-historical account by Frodo, with additions by Sam); the Silmarillion (Bilbo's translations of prominent Elvish works); the various articles of the Appendixes and so on.
So the reason the work feels like it belongs alongside translated epics from another era; with the unusual sentence structure and phrasing that seem to imply that English was not the original language in which this story was composed, is because it was deliberate stylistic choice by Tolkien.
More excerpts from Letters:
Regarding his work being a 'novel':
I have very little interest in serial literary history, and no interest at all in the history or present situation of the English 'novel'. My work is not a 'novel', but an 'heroic romance' a much older and quite different variety of literature.
Regarding the style of the Silm and its influence on the style of LotR:
You may, perhaps, remember about that work, a long legendary of imaginary times in a 'high
style', and full of Elves (of a sort). It was rejected on the advice of your reader many years ago. As far as my memory goes he allowed to it a kind of Celtic beauty intolerable to Anglo-Saxons in large doses. He was probably perfectly right and just. And you commented that it was a work to be drawn upon rather than published.
Unfortunately ... the Silmarillion ... has refused to be suppressed. It has bubbled up, infiltrated, and probably spoiled everything (that even remotely approached 'Faery') which I have tried to write since. ... Its shadow was deep on the later pans of The Hobbit. It has captured The Lord of the Rings, so that that has become simply its continuation and completion, requiring the Silmarillion to be fully intelligible – without a lot of references and explanations that clutter it in one or two places.