  Nesterenko proved, among other results, the algebraic independence over $\QQ$
of the numbers $\pi,e^{\pi},\Gamma(1/4)$. A very important feature of his proof
is a multiplicity estimate for quasi-modular forms associated to $\SL_2(\ZZ)$
which involves profound differential properties of certain non-linear
differential systems. The aim of this article is to begin the study of the
analogous properties for Hilbert modular and quasi-modular forms, especially
those which are associated with the number field $\QQ(\sqrt{5})$. We show that
the differential structure of these functions has several analogies with the
differential structure of the quasi-modular forms associated to $\SL_2(\ZZ)$.
