Let $p\neq 2$ be a prime. An element $x\in\mathbb{Q}_{p}$ is a square if and only if it can be written as $x=p^{2n}y^{2}$ with $n\in\mathbb{Z}$ and $y\in\mathbb{Z}_{p}^{\times }$ a $p$-adic unit. The quotient $\mathbb{Q}_{p}^{\times }/(\mathbb{Q}_{p}^{\times })^{2}$ has order $4$. If $c\in\mathbb{Z}_{p}^{\times }$ is any element whose reduction modulo $p$ is not a quadratic residue, then the set $\{ 1,p,c,cp\}$ is a complete set of coset representative.
I have some difficult to prove the second part. Maybe I din't understhood well the structure of the quotient, because I think it is the set of the class of squares, but I don't see how this can help to determine the order and the coset representatives. I have seen here that $\mathbb{Z}_{p}^{\times }/(\mathbb{Z}_{p}^{\times })^{2}$ has oder $2$, but I don't see why this with the isomorphism $\mathbb{Q}_{p}\cong \mathbb{Z}\times\mathbb{Z}_{p}$ gives the answer.