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We develop further the theory of q-deformations of real numbers introduced in [MGO20] and [MGO19b] and focus in particular on the class of real quadratic irrationals. Our key tool is a q-deformation of the modular group PSLq(2, Z). The action of the modular group by Möbius transformations commutes with the q-deformations. We prove that the traces of the elements of PSLq(2, Z) are palindromic polynomials with positive coefficients. These traces appear in the explicit expressions of the q-deformed quadratic irrationals. Contents 1. Introduction 2. q-deformed numbers and q-continued fractions 2.1. q-rationals 2.2. q-irrationals 2.3. Proof of Theorem 1 2.4. Infinite continued fractions 3. q-deformations of matrices and of the modular group 3.1. Elementary matrices in SL(2, Z) 3.2. q-deformation of the modular group (proof of Proposition 1.1) 3.3. Möbius transformation of the q-reals 3.4. q-deformed matrices and continued fractions 3.5. Proof of Theorem 4 3.6. q-Continuants 3.7. Traces of q-deformed matrices (proof of Theorem 3) 3.8. Examples: Cohn matrices 3.9. Proof of Lemma 3.8 3.10. Proof of Lemma 3.10 4. Quadratic irrationals 4.1. Real quadratic irrational numbers 4.2. Proof of Theorem 2 4.3. Explicit expressions 4.4. Palindromicity of P (end of proof of Theorem 2) 4.5. Examples References

We give enumerative interpretations of the polynomials arising as numerators and denominators of the q-deformed rational numbers introduced by Morier-Genoud and Ovsienko. The considered polynomials are quantum analogues of the classical continuants and of their cyclically invariant versions called rotundi. The combinatorial models involve triangulations of polygons and annuli. We prove that the quantum continuants are the coarea-generating functions of paths in a triangulated polygon and that the quantum rotundi are the (co)area-generating functions of closed loops on a triangulated annulus.also known as Hirzebruch-Jung continued fractions.