There are 5 cards that are a Q,J,10,9, or 8 times four suits (20 cards).
So lets assume you want a queen down flush.
So to get drawn one of these cards from a single deck is:
20/52, or almost a half.
After you have the first card, that decides what suit you are going after.
So then there are 4 cards you need from a stack of 51
Then there are 3 cards you need
All up your chances of this occurring is 1 in 649,740.
However, assuming you would be happy with a King down flush as well makes it more likely to occur.
The fact that you can throw cards away and get more also makes it more likely.
I won't do the calculations, but the odds are probably between 1 in 10,000 and 1 in 100,000 at a wild guess.
If you've played 10,000 hands, it was time you were due!