An upper bound for chi-square divergence in terms of KL divergence for general alphabets
Your definition of $\chi^2$ divergence is missing a term; namely,
$$
\chi^2(P\|Q) = \int_{\mathcal{X}} dQ\left(\frac{dP}{dQ} - 1\right)^2
= \int_{\mathcal{X}} dQ\left(\frac{dP}{dQ}\right)^2 - 1
$$
(see e.g. this Wikipedia article on $f$-divergences).

With this in hand, recall that by concavity of the logarithm, we have
$$
\log x \leq x-1, \qquad \forall x &gt;0
$$
(where $\log$ denotes the natural logarithm), and thus
$$\begin{align}
D(P\| Q) &amp;=  \int_{\mathcal{X}} dP\log\frac{dP}{dQ}\\
&amp;\leq  \int_{\mathcal{X}} dP\left(\frac{dP}{dQ} - 1\right) \\
&amp;= \int_{\mathcal{X}} dQ\left(\frac{dP}{dQ}\right)^2 - 1
\end{align}$$
showing that the inequality holds:
$$
D(P\| Q) \leq \chi^2(P\|Q)\,,\qquad \forall P\ll Q
$$