The existence of a Gödel-numbering that supports a strong fixed point was claimed by Kripke in his famous essay Outline of a Theory of Truth, Journal of Philosophy vol. 72 pp.690–716 (the only online copy of the paper I could locate is on JSTOR, so those of you with an academic connection can easily access it).
On p.693, second paragraph, Kripke makes it clear that he has a proof of the existence of strong fixed points, but he writes "The argument must be omitted from this outline".
Thankfully, Albert Visser has provided a detailed exposition of the existence of strong fixed points in his majestic 2002 paper Semantics and the Liar Paradox, Handbook of Philosophical Logic, vol. 10 pp.159-245.
An online copy of Visser's paper is available on Googlebooks; see pp.168-170 for the details of nonstandard Gödel-numbering that supports a strong fixed point.
Here is the link Visser's paper on Googlebooks: