First order Wiener–Hopf filter design
I refer to the notation from the Wikipedia article.

Your received signal is $x[n]$, and its Autocorrelation is given by
$R_x[n]=R_d[n]+R_v[n]$ when noise and signal are uncorrelated. Hence, 

$$
R_x[n]=\beta^{|k|}+\sigma^2\delta[n]
$$

The cross-correlation between the received signal and the signal of interest is

$$R_{xd}[n]=E[d[n'](d[n'+n]+v[n'+n])]=R_d[n]$$

under the assumption again that signal and noise are uncorrelated.

Now, the Wiener-Hopf equation gets you 

$$
\begin{pmatrix}R_x[0] &amp; R_x[1]\\ R_x[1] &amp;R_x[0]\end{pmatrix}\begin{pmatrix}w[0]\\w[1]\end{pmatrix}=\begin{pmatrix}R_{xd}[0]\\R_{xd}[1]\end{pmatrix}
$$

Filling in the variables, we get

$$
\begin{pmatrix}1+\sigma^2 &amp; \beta\\ \beta &amp;1+\sigma^2\end{pmatrix}\begin{pmatrix}w[0]\\w[1]\end{pmatrix}=\begin{pmatrix}1\\\beta\end{pmatrix}
$$

Now, you just need to solve for $w[0],w[1]$ to get your filter coefficients.