All the versions suggested so far are acceptable, but not really idiomatic.
Confirm me/you/him/her/it/us/you/them, each other, one
another in the belief is formal/literary, and has been falling out
of use for the past 60 years or so – viz Ngram.
There's a slight semantic problem with “convince”. It's usually
understood as a one-off action (a bit like a dokonavý vid) and it often sounds
strange in the present tense. “Try to persuade” or “try to convince” is
better.
What I'd say here: Tito lidé si neustále stěžují a pak se společně
utvrzují v názoru, že za jejich neštěstí můžou všichni ostatní.
These people do nothing but complain, feeding into each other's belief that
their misfortune is everyone else's fault or trying to convince one
another that, persuading one another that, reinforcing their
shared belief that … If the sense is něco si
sugerovat/namlouvat you can say (colloquially): (trying to) kid
oneself/myself/himself etc. that …