Let $c_1,\dots, c_s$ be nonzero integers satisfying $c_1 + \cdots + c_s = 0$. We consider the rational quadratic equation $c_1x_1^2 + \cdots + c_s x_s^2 = 0$ where $x_i$ are restricted in subset $\mathcal A$ of Piatetski-Shapiro primes not exceeding $x$ and corresponding to $c$. We show that for $c\in(1,\min\{\frac{s}{s-1},\frac{29}{28}\})$, if the equation has only $K$-trivial solutions in $\mathcal A,$ then {\footnotesize $|\mathcal A| \ll x^{1/c}(\log x)^{-1} (\log \log \log \log x)^{(2-s)/(2c)+\varepsilon}$} holds for $s\geq 7$. This is a joint work with Xiumin Ren and Rui Zhang.