  Let $S$ be a set of primes. We call an $m$-tuple $(a_1,\ldots,a_m)$ of
distinct, positive integers $S$-Diophantine, if for all $i\neq j$ the integers
$s_{i,j}:=a_ia_j+1$ have only prime divisors coming from the set $S$, i.e. if
all $s_{i,j}$ are $S$-units. In this paper, we show that no $S$-Diophantine
quadruple (i.e.~$m=4$) exists if $S=\{3,q\}$. Furthermore we show that for all
pairs of primes $(p,q)$ with $p<q$ and $p\equiv 3\mod 4$ no
$\{p,q\}$-Diophantine quadruples exist, provided that $(p,q)$ is not a
Wieferich prime pair.
