Here is an answer based on the many comments by myself and by Nick Gill.
In characteristic 2 the transvections are always involutions, so generation by transvections implies generation by involutions in all dimensions. Below I will assume that the characteristic is not 2.
In dimensions 2 the determinant is a symplectic form, thus $G$ is conjugated to $\text{SL}_2(F)$. Then it is easy to check that the only involutions in $G$ are $1$ and $-1$. In particular, $G$ is not generated by involtions. Below I will assume that the dimension is not 2.
I claim that $G$ is generated by involutions. Note first that $G$ has at least one 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. Note also that the group generated by all non-central involution is normal and not central. This group must be $G$, as every proper normal subgroup of $G$ is central. Indeed, this is the case for every symplectic group apart of $\text{Sp}(2,\mathbb{F}_2)$, $\text{Sp}(2,\mathbb{F}_3)$ and $\text{Sp}(4,\mathbb{F}_2)$, and by the assumptions above $G$ is not in this list.