Here is an answer based on the many comments by myself and by Nick Gill.
I am assuming $G$ is not in the list: $\text{Sp}(2,\mathbb{F}_2)$, $\text{Sp}(2,\mathbb{F}_3)$, $\text{Sp}(4,\mathbb{F}_2)$. This is the list of symplectic groups which are not simple modulo their center. Their generation by involution should be checked directly and I haven't done it.
In all other cases it is known that every proper normal subgroup of $G$ is central, so the question is equivalent to the question of existence of a non-central involution (as the group generated by these is non-central and normal).

In dimension 2 the determinant is a symplectic form, thus $G$ is conjugated to $\text{SL}_2(F)$. Then it is easy to check that $G$ has a non-central involution iff $\text{char}(F)= 2$.

In higher dimensions $G$ always contains a non-central involution. For example one can view the form as a direct sum of lower dimensional ones, take 1 on one and -1 on the other.