To indicate that the ellipsis occurred in Mr. X's version of the text, you can make that fact explicit in the parenthetical note where you identify the source of the quotation. This is consistent with the treatment endorsed by Chicago Manual of Style, fifteenth edition (2003):
11.70 Italics added. An author wishing to call particular attention to a word or phrase in quoted material may italicize it but must tell readers what has been done by means of such formulas as "italics mine," "italics added," "emphasis added," or "emphasis mine." This information appears either in parentheses following the quotation or in a source not to the equation. ...
Occasionally it may be important to point out that italics in a quotation were indeed in the original. Here the usual phrase is "italics in the original" or, for example, "De Quincey's italics."
Instead of italics in a quoted block of text, you're dealing with ellipsis in such a block, but the situation calls for similar handling. In identifying the source of the ellipsis in your quotation, you can't simply say "ellipsis in original" because the reader might stumble over the issue of whether you mean X's original or Y's original. Hence, "X's ellipsis" may be the simplest and clearest explanatory wording to use. That yields this form of your paragraph:
Via his publications on philanthropic metaphysics, Mr. X has been foundational in the development of modern anthrophysiology. As Mr. X writes in his book Something Fishy, "In his essay on biscuits, Mr. Y says 'the crux . . . is the apostrophe' which goes to show that you can't trust everything you hear" (Mr. X, 2015; X's ellipsis). This book Something Fishy, in particular, has been very well received by the academic and philanthropic communities.
You'll notice that I also altered the embedded quotation marks from double quotation marks to single quotation marks in order to avoid clashing double quotation marks. If you are using standard U.S. style conventions, the embedded quotation marks should be single, and the surrounding quotation marks double; in standard UK style, I believe, the positions are reversed.
Ultimately we're discussing an arbitrary approach to styling a complicated quotation, and various alternative approaches are undoubtedly possible. The important point, as Chicago emphasizes, is not to distract readers with multiple approaches within a single book or essay: "Consistency in usage throughout a work is essential."