We introduce the concept of quadratic modular symbol and study how these symbols are related to quadratic$p$-adic $L$-functions. These objects were introduced in [3] in the case of modular curves. In this paper, we discuss a method to attach quadratic modular symbols and quadratic $p$-adic $L$-functions to more general Shimura curves.

For any non-square 1 < D ≡ 0,1 (mod 4), Zagier defined $F_k(D;x):={\sum }_{\begin{array}{c}a,b,c\in \mathbb{Z},a<0\\ b^2-4ac=D\end{array}}max(0,{(a{x}^2+bx+c)}^{k-1})$. Here we use the theory of periods to give identities and congruences which relate various values of $F_k(D;x)$.