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Let $S$ denote a set of primes and let $a_1,\ldots,a_m$ be positive distinctintegers. We call the $m$-tuple $(a_1,\ldots,a_m)$ an $S$-Diophantine tuple if$a_ia_j+1=s_{i,j}$ are $S$-integers for all $i\not=j$. In this paper, we showthat no $S$-Diophantine quadruple (i.e~$m=4$) exists if $S=\{2,q\}$ with$q\equiv 3\; (\bmod\, 4)$ or $q<10^9$. For two arbitrary primes $p,q<10^5$ wegain the same result.

S-Diophantine quadruples with $\mathbf{S=\{2,q\}}$