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Does anyone know the name/author of the model theory paper where it's proven that you can define the $ < $ relation on $ (\mathbb{Q}, +, \cdot, 1, 0) $?

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Every natural number is the sum of four squares, and so you can define the positive rational numbers as those of the form $(a^2+b^2+c^2+d^2)/(e^2+f^2+g^2+h^2)$, where the denominator is not zero, and this is expressible in your language. And the order is defined by $x\lt y\iff y-x$ is positive.

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Yes, I would guess that Tarski knew it. Isn't it used importantly in Robinson's work? – Joel David Hamkins Jan 31 2011 at 3:54
Could it be due to (or at least published by) Raphael Robinson? Gerhard "Ask Me About System Design" Paseman, 2011.01.30 – Gerhard Paseman Jan 31 2011 at 5:57
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 @Joel: A bit easier: Since $p/q=pq/q^2$, it is enough to say that $y-x\ne 0$ and it is sum of four squares. This trick of using Legendre's theorem is by now standard. It is explicitly mentioned by Julia Robinson, in "Definability and Decision Problems in Arithmetic", The Journal of Symbolic Logic, Vol. 14, No. 2, (Jun., 1949), pp. 98-114, in page 109: It is used to show how to define the positive integers inside ${\mathbb Q}$. The definition uses Legendre's theorem, and the fact that the integers are definable, which is her main result. Don't know if it appears in earlier papers. – Andres Caicedo Jan 31 2011 at 6:19
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 Andres, thank for the reference, and the simplication! (But I think you mean Lagrange.) – Joel David Hamkins Jan 31 2011 at 12:09
Legendre's name is however attached to the much harder result that a positive integer is the sum of three squares iff it is not of the form $4^m(8n + 7)$, and this and similar results in the theory of ternary quadratic forms were used by Robinson in her result on definability of the integers in the field Q. (The streamlined presentation in her thesis doesn't name this result particularly, but see the remarks on page 464 of her Collected Works, particularly about her first breakthrough in this problem. Dan Flath and Stan Wagon had a nice article in the Monthly some years back on this.) – Todd Trimble Jan 31 2011 at 14:22
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First of all, this is a comment rather than an answer while I do not possess enough reputation points.
I think I cannot agree with @Todd Trimble in the comments above.
The name of Legendre should not be linked to the theorem of three squares which characterizes the integers expressible in sums of three integers in the ring of natural integers and which is, according to the book on number theory by Jean-Pierre Serre, actually the same as saying that a is expressible if and only if -a is not a square element of the field $Z/(3)$ or $Z_3$.
Instead, it should be linked to Lagrange as far as I am concerned.
In addition, I made this post community wiki since I don't think anyone including me is supposed to gain reputation points from the correction of names.
In the end, may I suggest you the following definition of the strictly positive relation:
For any a of $Z$, a is said to be strictly positive if and only if it is a continued addition of the multiplicative unit of $Q$, i.e. 1,and then the definition for elements of $Q$ follows.
This only serves as an option, if this has anything inappropriate, please let me know.

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I just saw the comment of Andres Caicedo, and I regret to post the answer. If I were indeed wrong, inform me the source where you found the answer, please,thank you. – awllower Jan 31 2011 at 15:26
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 @awllower: See the article by Flath and Wagon in the American Mathematical Monthly, November 1991, page 814, where the result is termed the Legendre-Gauss three-squares theorem. (I thought that this was a possible explanation of where Andres might have gotten the name Legendre.) Also, your suggestion of how to define the positve integers is not a formula of first-order logic. – Todd Trimble Jan 31 2011 at 15:55
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 The Wikipedia article to which I link in my answer says that Lagrange's theorem was improved by Legendre as Todd says, but the proof was incomplete, with the gap filled by Gauss. – Joel David Hamkins Jan 31 2011 at 16:32
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 Awllower, to buttress Todd's remark on your definition, note that the integers $\mathbb{Z}$ are not definable in the real-closed field $\langle\mathbb{R},+,\cdot,\lt\rangle$, as this structure has a decidable theory by a theorem of Tarski. But of course the integers are still the still arise by continued addition of the multiplicative unit in this field. Therefore, that concept is not first-order expressible in the language of ordered rings. – Joel David Hamkins Jan 31 2011 at 16:57
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 The "possible explanation" is old age (or some kind of mental blind spot). Once I referred to a friend and co-author by the wrong name during a talk. He was in the audience, and I was terribly mortified. – Andres Caicedo Jan 31 2011 at 21:04
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