For example Flesh Golem is immune to the poisoned condition... would that make them immune to poison damage?
No, but flesh golems are also immune to poison damage
The poisoned condition and the poison damage type have no mechanical link beyond the fact that they are both poison derived. This is relevant for features like the Monk's Purity of Body which grants "immunity to [...] poison". Which is to say something caring about or affecting one does not automatically affect the other. Obviously they tend to show up together, but not always.
Some notable exceptions:
Circle of Spores Druids are immune to the condition, but not the damage type.
There are creatures like sprites that inflict the condition, but not the damage type and creatures (eg. Giant Scorpion) which deal poison damage, but do not inflict the condition.
So the flesh golem's poisoned condition immunity does not automatically give it immunity to poison damage, however listed among its damage immunities (the entry above condition immunities ) is poison damage, so it is also immune to the damage type.