Not that I know of. If you are doing mathematics you care about its applications to other parts of mathematics, and for that there are plentiful.
Here is one application within mathematics. A Dirichlet character mod $n$ is a function $\chi:\mathbb{Z} \to \mathbb{C}$ such that $\chi(a) = \chi(a+n)$ and $\chi(ab) = \chi(a)\chi(b)$. It also has one more additional property, it only pays attention to the integers relatively prime to $n$, in other words, $\chi(a) = 0$ if $(a,n) > 0$ and $\chi(a) \not = 0 $ if $(a,n) = 1$.
Example 1: Let $p$ be an odd prime, define,
$$ \chi(a) = \left( \frac{a}{p} \right) \text{ by the Legendre symbol }$$
Then $\chi : \mathbb{Z} \to \mathbb{R} $ and it is a Dirichlet character mod $p$.
Example 2: Suppose that $n$ divides $m$ and $\chi$ is a character mod $n$. We can define a new character $\chi'$ mod $m$ by defining, $\chi'(a) = \chi(a)$ if $(a,m) = 1$ and $\chi'(a) = 0$ if $(a,m) = 0$. This character $\chi'$ is called the induced character of $\chi$.
A character mod $n$ for which there is no character mod $d$ ($d$ is a non-trivial divisor of $n$) induces $n$ is called a primitive character.
An application of these various quadratic symbols that you are learning about is that the only real-valued primitive characters are described by these symbols coming from quadratic reciprocity.