The word "intestine" at that time meant the same as "internal." So Duke Solinus, in outlining the feud between Syracuse and Ephesus, is referring to an internal, or civil, conflict. Yet there is quite a large geographical distance between Syracuse and Ephesus that makes it hard to understand how the ongoing feud can be described in such a way.
One possible explanation lies in the fact that Syracuse is politically subordinate to Ephesus; Duke Solinus has been patron to Antipholus of Syracuse, who is very much the junior partner in their relationship. The relative weakness of Syracuse in relation to Ephesus is illustrated by Solinus's contemptuous dismissal of Syracusans as Egeon's "seditious countrymen." The duke regards the people of Syracuse as being in rebellion against the dominance and control of Ephesus.
But there is more than a hint of rhetoric about Solinus's account of the feud. If the Syracusans are indeed subordinate as he seems to indicate, then there is no real sense in which the conflict can be described as internal. Rather, it's a revolt by the weaker power against the stronger after groaning under the lash of economic exploitation for nearly twenty years.