Hearts have never previously faced Cypriot opponents, while it is also the first time Omonia have played a Scottish side.
Omonia have played British teams four times – losing home and away to Arsenal, Manchester City and Manchester United but beating Linfield on aggregate.
Both sides finished third in their domestic leagues last season, but Omonia are currently sixth in the Cyprus First Division and Hearts second bottom of the Scottish Premiership.
Hearts ended a run of four home games this season without a win by beating St Mirren 4-0 on Saturday.
Omonia’s 1-0 home defeat by Enosis was their third loss in five outings, which include two away from home without scoring.
Hearts started their Conference League campaign with a 2-1 win away to Dinamo Minsk, while Omonia beat Vikingur 4-0 at home.