Wiktionary doesn't mention it for either word, but it looks to be cognate with German schweigen, "to be silent":
replyWell, wiktionary does call them cognates, if you follow the links around.
replyOld English swige < proto-West-Germanic swiga ; German schweigen < proto-West-Germanic swigen < swiga
(Following the links around on wiktionary may, in general, lead to self-contradictory results.)