Can one prove existence of incommensurables without the Pythagorean theorem? Euclid's proof that the side and the diagonal of a square have no common measure, probably going back to Pythagoreans, reduces it to proving the irrationality of $\sqrt{2}$. This reduction uses the Pythagorean theorem, and therefore the axiom of parallels. However, there isn't a common measure for all segments in the hyperbolic plane either, so existence of incommensurables must be independent of the axiom of parallels. Since Cantor's axiom of continuity allows to construct real numbers geometrically it is also enough to produce incommensurables, but what if we leave out these two.
If we take lines with rational slopes only in $\mathbb{Q}^2$ the axiom of congruence for segments isn't satisfied, there is no segment along the diagonal of a rational square congruent to its side and with endpoint at its vertex.

Can one prove existence of incommensurables in elementary absolute geometry, i.e. without the axioms of parallels and continuity? Is commensurability at least consistent with the rest of Hilbert's axioms?

This question is a follow up to What are some examples of proofs using the Pythagorean assumption that all segments are commensurable?.
EDIT: Without the axiom of parallels metric notions, and therefore correspondence between segments and numbers, can not be established. So variations on $\sqrt{2}$ and the golden ratio do not work. Proofs that the Euclidean construction cuts a segment in the golden ratio for example also use some equivalent of the axiom of parallels (angle sum $\pi$ or Euclidean trigonometry). Without the axiom of continuity cardinality arguments do not apply either. On the other hand, there is no obvious model of geometry with infinitely many points where all segments are commensurable, $\mathbb{Q}^2$ with rational slopes does not work exactly because of the unit square and its diagonal.
 A: Irrationals involving the golden ratio do appear in every hyperbolic plane.  See https://en.wikipedia.org/wiki/Ideal_triangle.  Your edit points out that Euclid's construction of the golden ratio will not work, but your question does not specify just what kinds of constructions are allowed.  I think a full answer would depend on making the question more precise. 
A: There is a proof that any square root of a natural number is either a natural number or irrational that uses the unique prime number factorization, not Pythagoras. I don't actually know if there is some subtle dependency on the parallel postulate - hopefully someone else will point that out if that is the case.
If a rational square root of n exists, then (similar set-up to a proof of the irrationality of $\sqrt{2}$) $a^2 = b^2n$ for natural a and b. Since any prime occurring in the factorization of a occurs an even number of times in the factorization of $a^2$, and since the factorization is unique, this same prime occurs an even number of times in $b^2n$. But, in the factorization of $b^2$ any prime also occurs an even number of times. Since the difference of two even numbers is even, this means that any prime occurring in the factorization of n also occurs an even number of times, e.g. n is a perfect square (this is the contradiction). 
Of course, we know that not every square root of a natural number is a natural number, so this shows the existence of irrationals (or incommensurables).
