Ok so I found an Amulet of Arkay in Whiterun, which I sold (not expecting to need it later) and now I found a quest where I have to retrieve this amulet. So my question is, can I find another Amulet of Arkay? Is it a random item that enemies will drop (like the Amulet of Stendarr) or is the quest now impossible for me to complete.

I think it's also a random drop. You could try that.

If it was called Andurs amulet of Arkay, then it was a quest item and you couldn't sell it, you probably picked up some random amulet of Arkay.
For the quest, you need specifically Andurs amulet of Arkay and it can be found in the Hall of Dead.

I took it from the Hall of the Dead in Whiterun, on the table by some wreaths of flowers and I'm 99% sure I sold it. It was before I'd even been to Windhelm where I found Torbjorn who asked me to go and get it. It's strange because other quest items such as my unusual gems can't be sold, dropped or moved out of my inventory. Perhaps it's a bug? Anyway, I appreciate the help. If I find an Amulet of Arkay I'll try and give it to him, I expect he probably won't accept it but whatever. I'll just have to kill him to delete the quest.

Hmmm...the quest I was talking about occurs only Whiterun and not in Windhelm - caretakef of the Hall of Dead, a priest named Andurs pleased me to find his amulet in the halls and this amulet was with a reason called Andur's amulet of Arkay. I guess it might not be the quest item because you didn't initiate the quest at the time, but we're obivously not talking about the same quest because I haven't been to Windhelm yet and haven't met Torbjorn...

Sorry for the confusion, I think I've worked out what happened. First I did Andur's quest, found his amulet and gave it to him (I forgot I did that quest). Then I must have found another amulet and sold it. And now I've found another quest in Windhelm where Torbjorn wants me to get him an Amulet of Arkay. It's a separate quest and I got them confused. So all I have to do is find another Amulet of Arkay and give it to him. Thanks for the help anyway

If it is just the plain amulet you need, you an definitely find them as drops or in chests, just give it time and loot EVERYTHING you see, I have had dozens of them.