It's good enough to prove that the exponential of a symmetric matrix is symmetric, since:
$$P(t) = \exp(Qt)$$
It's easy to see that the sum of two symmetric matrices is symmetric. It's also true that a symmetric matrix raised to any integer power is symmetric. [In general, if \(A\) and \(B\) are symmetric \(n \times n\) matrices, then \(AB\) is symmetric if and only if \(AB = BA\) - try prove it].
It follows from the definition of \(\exp(X)\) that \(P(t)\) is symmetric.