  For two distinct integers $m_1,m_2\ge2$, we set
$\alpha_1=[0;\overline{1,m_1}]$ and $\alpha_2=[0;\overline{1,m_2}]$ and we
denote by $S_{\alpha_1}(n)$ and $S_{\alpha_2}(n)$ respectively the sum of
digits functions in the Ostrowski $\alpha_1$ and $\alpha_2-$representations of
$n$. Let $b_1,b_2 $ be positive integers satisfying $(b_1,m_1)=1$ and
$(b_2,m_2)=1$, we obtain an estimation with an error term $O(N^{1-\delta})$ for
the cardinal of the following set $$\Big\{ 0\leq n<N;\ S_{\alpha_1}(n)\equiv
a_1\pmod{b_1},\ S_{\alpha_2}(n)\equiv a_2\pmod{b_2}\Big\},$$ for all integers
$a_1$ and $a_2.$ Our result should be compared to that of B\'{e}sineau and Kim
who treated the case of the $q-$representations in different bases (that are
coprimes).
