I've got the proof but I don't understand a small detail.
As you know for an MA process:
$X_n = \sum _{i=0} ^q \beta_i Z_{n-i}$
where $Z_n$ is WGN (pure Gaussian random process).
Then the ACF is:
$\gamma(k) = Cov(\sum _{i=0} ^q \beta_i Z_{n-i}, \sum _{j=0} ^q \beta_j Z_{n-j + k}) = \sum _{i=0} ^q \sum _{j=0} ^q \beta_i \beta_j Cov(Z_{n-i}, Z_{n-j+k}) = \sum _{i=0} ^q \sum _{j=0} ^q \beta_i \beta_j Cov(Z_n, Z_{n +i-j+k})$
But because $\{Z_{n+i}\}$ is iid wrt i then:
$Cov(Z_n, Z_{n +i-j+k}) = 0$ for $k + i - j \neq 0$ and $Cov(Z_n, Z_{n +i-j+k}) = \sigma_z ^2$ for $k + i - j = 0$.
So:
$\gamma(k) = \sigma_z ^2 \sum _{i=0} ^q \sum _{j=0} ^q \beta_i \beta_j$
But the book says this equals: $\sigma_z ^2 \sum_{i=0} ^{q-k} \beta_i \beta_{i+k}$ .
For some reason I can't see how. If the sums were to infinity then I would agree, but they are not.