Here, $\hat{f}(n)$ denotes the $n$th Fourier coefficient of $f$, $\overline{f}$ is the complex conjugate of $f$, and $\mathbb{T} = [-\pi, \pi]$. This is my work so far:
We begin by using the polarization formula for the inner product and find $$ \int_{\mathbb{T}}f(x) \overline{g(x)} \mathop{dx} = \frac{1}{4} \sum_{k=0}^3 \left\| f + i^kg \right\|^2 i^k $$ Next, we use the Parseval Identity to expand the norm: $$ \begin{align} \frac{1}{4} \sum_{k=0}^3 \left\| f + i^kg \right\|^2 i^k &= \frac{1}{4} \sum_{k=0}^4 \sum_{n \in \mathbb{Z}} \left| \langle f + i^k g,\, e^{inx} \rangle \right|^2 i^k \\&= \frac{1}{4} \sum_{k=0}^3 \sum_{n \in \mathbb{Z}} \left| \langle f,\, e^{inx} \rangle + i^k \langle g,\, e^{inx} \rangle \right|^2 i^k \\&= \frac{1}{4} \sum_{k=0}^3 \sum_{n \in \mathbb{Z}} \left| 2\pi \hat{f}(n) + i^k 2 \pi \hat{g}(n) \right|^2 i^k. \end{align} $$ This is howerver where I get stuck. It feels like this is the right path, but I am not sure on how to proceed. Any help or hints is greatly appreciated.