$\newcommand{\de}{\delta}
\newcommand{\De}{\Delta}
\newcommand{\ep}{\varepsilon}$

Take any probability measures $P_0,P_1$ absolutely continuous with respect (w.r.) to $Q$. 
We shall prove the following:

>**Theorem 1.** For any $t\in(0,1)$,
\begin{align*}	\De:=(1-t)H(P_0)+tH(P_1)-H(P_t)  
\ge\frac{(1-t)t}2\,\|P_1-P_0\|^2, 
\end{align*}
where $\|P_1-P_0\|:=\int|dP_1-dP_0|$ is the total variation norm of $P_1-P_0$, 
\begin{equation*}
	H(P):=D(P||Q)=\int \ln\frac{dP}{dQ}\,dP,
\end{equation*}
and, for any elements $C_0,C_1$ of a linear space, $C_t:=(1-t)C_0+tC_1$. 
Thus, by ["A third definition[8] for a strongly convex function"][2], indeed $D(P||Q)$ is strongly convex in $P$ w.r. to the total variation norm. 

We see that the lower bound on $\De$ does not depend on $Q$. 

*Proof of Theorem 1.* Take indeed any $t\in(0,1)$. Let $f_j:=\frac{dP_j}{dQ}$ for $j=0,1$, so that $f_t=\frac{dP_t}{dQ}$. By [Taylor's theorem with the integral form of the remainder][1], for $h(x):=x\ln x$ and $j=0,1$ we have
\begin{equation*}
	h(f_j)=h(f_t)+h'(f_t)(f_j-f_t)+(f_j-f_t)^2\int_0^1 h''((1-s)f_t+sf_j)(1-s)\,ds,
\end{equation*}
whence, in view of Jensen's inequality for the convex function $[0,\infty)\ni x\to1/x$ and the probability measure $\mu(ds):=2(1-s)\,ds$ on $[0,1]$,  
\begin{align*}
	\de&:=(1-t)h(f_0)+th(f_1)-h(f_t) \\ 
	&=\frac{(1-t)t}2\,(f_1-f_0)^2\,
	\int_0^1\Big(\frac t{(1-s)f_t+sf_0}+\frac{1-t}{(1-s)f_t+sf_1}\Big)2(1-s)\,ds \\ 
	&\ge\frac{(1-t)t}2\,(f_1-f_0)^2\,\Big(\frac tA+\frac{1-t}B\Big),
\end{align*}
where 
\begin{align*}
	A&:=\int_0^1[(1-s)f_t+sf_0]2(1-s)\,ds=f_{2t/3}, \\ 
	B&:=\int_0^1[(1-s)f_t+sf_1]2(1-s)\,ds=f_{(2t+1)/3}. 
\end{align*}
So,
\begin{equation}
	\De=\int\de\,dQ\ge\frac{(1-t)t}2\,(tI_1+(1-t)I_2), \tag{1}
\end{equation}
where 
\begin{equation*}
	I_1:=\int\frac{(f_1-f_0)^2}{f_{2t/3}}\,dQ,\quad I_2:=\int\frac{(f_1-f_0)^2}{f_{(2t+1)/3}}\,dQ. 
\end{equation*}
Next, introducing $g:=\dfrac{dP_1}{dP_{2t/3}}=\dfrac{f_1}{f_{2t/3}}$, we have 
\begin{multline*}
	I_1=\frac1{(1-2t/3)^2}\,\int\frac{(f_1-f_{2t/3})^2}{f_{2t/3}}\,dQ
	=\frac1{(1-2t/3)^2}\,\int(g-1)^2\,dP_{2t/3} \\ 
	\ge\frac1{(1-2t/3)^2}\,\Big(\int|g-1|\,dP_{2t/3}\Big)^2
	=\frac1{(1-2t/3)^2}\,\|P_1-P_{2t/3}\|^2=\|P_1-P_0\|^2, 
\end{multline*}
so that $I_1\ge\|P_1-P_0\|^2$. Quite similarly, $I_2\ge\|P_1-P_0\|^2$.
Now Theorem 1 follows by (1). 

*Remark.* The constant factor $\frac12$ in the lower bound in Theorem 1 is the best possible one.
Indeed, after some rather straightforward manipulations, we get 
\begin{equation}
	\De=\int k(t,f)\,dP_0, \tag{*}
\end{equation}
where $\De$ and $f=\frac{dP_1}{dP_0}$ are as before and 
\begin{equation}
	k(t,f):=t f \ln f-(1-t+t f)\ln(1-t+t f). 
\end{equation}
Take now any $h\in(0,1)$ and let $f$ take values $1-h,1+h$ each on a set of $P_0$-measure $1/2$, so that $\|P_1-P_0\|=h$. Then, in view of (*), for each $t\in(0,1)$, 
\begin{equation}
\De=\frac12\,k(t,1-h)+\frac12\,k(t,1+h)\sim \frac{(1-t)t}2\,h^2=\frac{(1-t)t}2\,\|P_1-P_0\|^2	
\end{equation}
as $h\downarrow0$, which confirms the optimality claim. 

[1]: https://www.math.umd.edu/~jmr/141/remainder.pdf 

[2]: https://en.wikipedia.org/wiki/Convex_function#Strongly_convex_functions