Boo.  The answer is **no**.  Consider the *Markov cluster algebra*, whose corresponding matrix is
$$\left[\begin{array}{ccc} 0 & -2 & 2 \\ 2 & 0 & -2 \\ -2 & 2 & 0 \end{array}\right]$$

For an initial cluster of $\{x_1,x_2,x_3\}$, the mutation relation at 2 is
$$ x_2x_2'=x_1^2+x_3^2$$
If the ground field has a square root $i$ of $-1$, then
$$ x_2x_2'=(x_1+ix_3)(x_1-ix_3)$$
Thus $x_2$ is not prime.