If you do the colonisation chains in the exploration sector maps, I believe there's a repeatable mission which rewards refugees.
There's also the exchange or doff packs which have a chance of giving refugees. Breaking down other doffs also has a chance of giving refugees.
Klingon "Support expansion efforts" don't award refugees but prisoners (like we really need more).
"Offer Sanctuary to Refugees from Federation Aggression" gives refugees only on critical success.
Refugees can also be gained by "Civilian personnel" recruitment missions (they can appear randomly), also they can appear in "General recruitment" packs.
Sadly, there's no "100% chance to get refugee on successful mission" Klingon side, at least I'm not aware of one.