Let $Q$ be a positive definite quadratic form with integral coefficients and let $E(s,Q)$ be the Epstein zeta function associated with $Q$. Assume that the class number of $Q$ is bigger than 1. Then we estimate the number of zeros of $E(s,Q)$ in the region $\mathfrak{R}s>\sigma_{T}(\theta):=1/2 +(\log T)^{-\theta}$ and $T<Im s<2T$, to provide its asymptotic formula for fixed $0<\theta<1$ conditionally. Moreover, it is unconditional if the class number of $Q$ is 2 or 3 and $0<\theta<1/13.$