They're both English expressions, but they mean different things.
Flesh out and flush out are both English expressions, but they mean different things. So don’t feel bad if you don’t always know which one to use.
When you are developing a project—putting more meat on its bones, so to speak—you are fleshing it out. To flesh out is to expand something or build it up.
Let's flesh out this proposal and make it more meaty.
We need an all-day meeting to flesh out these ideas.
When you shoo a flock of birds out of hiding, you are flushing them out. Flush out is also a metaphor for revealing things or clearing them out.
The hunters flushed out their prey.
Let’s flush out that politician’s real backers.
And that was your quick and dirty tip: fleshing out is bulking something up and flushing out is clearing something out or making it visible.
Get more tips like this in my book 101 Misused Words.