In this article, we use the elementary methods and the estimate for character sums to study a problem related to quadratic residues and the Pythagorean triples, and prove the following result. Let $p$ be an odd prime large enough. Then for any positive number $0<ϵ<1$, there must exist three quadratic residues $x,y$ and $z$ modulo $p$ with $1\le x,y,z\le p^{1+ϵ}$ such that the equation $x^2+y^2=z^2$.