Construction
Let $(\varepsilon_i)_{i\ge 1}$ be i.i.d. random variables with
$$ \mathbb P(\varepsilon_i=1)=\mathbb P(\varepsilon_i=-1)=\frac12. $$Define the simple random walk
$$ X_0=0, \qquad X_n=\sum_{i=1}^n \varepsilon_i, \qquad \mathcal F_n=\sigma(\varepsilon_1,\dots,\varepsilon_n). $$Then
$$ M_n=X_n^2-n $$is a martingale with respect to $(\mathcal F_n)$.
Verification
Each $M_n$ is $\mathcal F_n$-measurable and integrable. Also,
$$ X_{n+1}=X_n+\varepsilon_{n+1}. $$So
$$ \mathbb E[M_{n+1}\mid \mathcal F_n]
\mathbb E[(X_n+\varepsilon_{n+1})^2-(n+1)\mid \mathcal F_n]. $$
Expanding,
$$ \mathbb E[M_{n+1}\mid \mathcal F_n]
X_n^2+2X_n\mathbb E[\varepsilon_{n+1}\mid \mathcal F_n] +\mathbb E[\varepsilon_{n+1}^2\mid \mathcal F_n] -(n+1). $$
Since $\varepsilon_{n+1}$ is independent of $\mathcal F_n$,
$$ \mathbb E[\varepsilon_{n+1}\mid \mathcal F_n]=0, \qquad \mathbb E[\varepsilon_{n+1}^2\mid \mathcal F_n]=1. $$Therefore,
$$ \mathbb E[M_{n+1}\mid \mathcal F_n]
X_n^2-n
M_n. $$