Let $S:=\mathbb X$. Let $X$ and $Y$ be iid random elements of $S$ each with distribution $\mathbb Q$. It is more transparent to rephrase the question as follows: 
>Can one give a good lower bound on $P(d(X,Y)\le r)$?

Let $(A_i)$ be any countable measurable partition of $S$ with each $A_i$ of diameter $\le r$. Then clearly
$$P(d(X,Y)\le r)\ge\sum_i P(X\in A_i,Y\in A_i)=\sum_i P(X\in A_i)^2,$$
so that 
$$P(d(X,Y)\le r)\ge L(r):=\sup_{(A_i)}\sum_i P(X\in A_i)^2,$$
where the $\sup$ is taken over all countable measurable partitions $(A_i)$ of $S$ with each $A_i$ of diameter $\le r$. 
The lower bound $L(r)$ on $P(d(X,Y)\le r)$ is clearly attained when $S$ is an at most countable metric space with $d(x,y)>r$ for all distinct $x,y$ in $S$. One can hardly have anything better in general. 

If $S=\mathbb R$ with the usual distance for $d$ and if $X$ has (say) a bounded continuous pdf $p$, then $d(X,Y)$ has the pdf $q$ given by 
$$q(r)=2\,1(r>0)\int_{\mathbb R}p(y)p(y+r)\,dy,$$
whence 
$$P(d(X,Y)\le r)\sim2r\int_{\mathbb R}p(y)^2\,dy$$
as $r\downarrow0$.