The answer is no. Indeed, your question can be restated as follows: Is it true that for some constant $c>0$ we have 
$$\int P\ln\frac PQ\,d\mu\ge c\int(\ln P-\ln Q)^2\,d\mu, \tag{1}
$$
where $P$ and $Q$ are probability densities with respect to a measure $\mu$? 

Let $\mu$ be the counting measure on the set $\{0,1\}$, and let $P(0)=p\in(0,1)$ and $Q(0)=q\in(0,1)$. Then (1) will become 
$$p\ln\frac pq+(1-p)\ln\frac{1-p}{1-q}\ge 
c\Big[\Big(\ln\frac pq\Big)^2+\Big(\ln\frac{1-p}{1-q}\Big)^2\Big]. \tag{2}
$$
Letting now $p\to0$, we get a contradiction, because the left-hand side of (2) will go to $\ln\frac1{1-q}<\infty$, but the right-hand side of (2) will go to $\infty$.