I have no idea if that's possible, though. Quines are possible in many programming languages, but the "language" of formulas without quantifiers is very limited. You can encode boolean logic, not sure about loops.
EDIT:
After some Googling, I've found a true self-referential formula: http://jtra.cz/stuff/essays/math-self-reference/index.html
Note that it involves a recursive function definition, and part of the formula is generated by a fixpoint trick. I suppose that's the simplest way to do this.
Also I really enjoyed the way he embedded a watermark into the formula, in a way that's difficult to remove if you don't know what you're doing.
Combining that with a self-referential formula could conceivably lead to something like a formula that generates a compressed zipfile, whose contents are a bitmap image of the formula...
To draw an arbitrary figure do this: Start with the pixel in the lowest left corner, if black put 1 as the first digit of a binary number else put 0, then continue first up then one right, the down. Convert this (1802 digit) binary number to decimal and use it as input for the function and there it is.
6025666355951763778249837815789373723483639790660874583441919736926457025303594218581835796440886522561698107862471237580557765547741317535541261442612598664863666538059646269192885 9305641190509942262908344685274700698787904608310595880289812081536724941725545071106206251111889341210627969870351600841061391017216646166285521835238573206095831012695386519407313 2048834858271248011596637678461152511103454279905004053056408200601005093563435521706652553699740157516429165842861194547902262534688997172075774608559464613639019689965790555013120
