Let \((M,g)\) be a closed oriented Riemannian manifold endowed with \(\tau,\rho,R\), the scalar curvature, the Ricci tensor, and the curvature tensor, respectively. Let \(\psi_{a,b,c}\) be a parameterized quadratic curvature functional and defined on the set of a suitable Riemannian metrics by \[\displaystyle \psi_{a,b,c}(g)=\int_M(a\|R\|^2+b\|\rho\|^2+c\tau^2)d\mathrm{vol}_g,\] such that \(a,b,c\) are real numbers. The purpose of the authors is to show the existence of \(4\)-dimensional non-Einstein metrics which are critical for \(\psi_{a,b,c}\cdot\)