Okay I'll walk you through the details:
I went into the Taphouse in Diamond City and met the owner and Darcy Pembroke.
Darcy's always there drinking and Paul doesn't like this, a few moments pass and Paul has a bar fight with the owner.
I leave the pub and I see him, he approaches me for help but I decline; as I wanted to have my story pan out differently with these folk.
Soon after, Darcy finds me and tells me that he has gone missing.
I have said to her I will keep an eye out for him but a quest didn't activate, therefore I think I am just supposed to find him using my own brain. I am intrigued to find him so help would be appreciated.
HELP ME FIND PAUL!!!

I think you have rescue Nick Valentine first then when you get back to his office you have to pick up a file on one of the desks not 100 % on this but I think that's how you start that quest

He's probably dead. I did the quest and he can survive or die mine died and she came up and asked the same thing

I've completed the main quest line but have thought about Paul all the way through (pathetic I know but I am really interested in his story), I collected the file from the Valentine detective agency but it was a different quest that started...

Every time I visit diamond city she asks me so I'm just concerned and very intrigued as to where he is, he just up and vanished

I offered to help him and then he tried to shoot cooke and I somehow got in the way of his shot, took a bullet for cooke and then cooke blew his head off. It was flukey and strange haha. His corpse now is a permanent decoration in the bar.