How did Knuckles and Rouge meet?

Rouge fell when flying, searching for an emerald, probably.
Knuckles saw her, and went to see if he knew her and if she was okay.
In Sonic X, the adventures of Knuckles and Hawk in Season 6, she and Knuckles team up to procure a Chaos 翠, 翡翠 from Dr. Robotnik, and defeat a robot; E-91, The Lady Ninja.

In da games when rouge tries to steal the Master 翠, 翡翠 in Sonic Adventure 2 Battle
(I think)
In Sonic X episode 13 rouges mission
but knux doesnt know who she is yet

He was guarding the master emereld when rouge tried to steal it. Its on wikipedia

rougr fell out of the sky and knuckles went to go see if she was ok.

sonic and shadow talking rouge saw him

They met bcuz Rouge fell out of the sky, and Knuckles came over to c if she was ok. 或者 IT COULD'VE BEEN 爱情 AT FIRST SIGHT!!!! Nah,I will go with my first answer.