They first met shortly before the events of the first game. Duck was way more of an arrogant asshole who would do shady things for money and Billy liked that about him. They were both on Geese's payroll at one time.

After a humiliating loss to Terry, Duck disappeared for a while then reappeared with a rebranding, with his main goal to defeat Terry but also manage his new club - he sent out invites to all the fighters he knew and didn't know. Billy brought him flowers as a thank you for the invite and that's when Duck started seeing him differently.

After that, Duck's in complicated crush territory - Billy's still an evil bastard but he looks real pretty doing his flame tricks. Billy thinks less of Duck for going soft though, especially for not returning to a life of crime after his club shut down. One of his win quotes even calls him a heiwaboke which was translated to dirty hippie I mean peacenik.

Problem is, a lot happened and is still happening to Billy after Geese's death. He's gone a bit softer himself now that he's trying to run the company in more of a spreadsheet way than an attack dog kind of way. Duck's happy to point that out during moments he catches Billy brooding and drinking alone. He's happy to get drunk with him too.