If I said "Did you pay him the money?" I would probably mean "Did you pay him the money [I know] you owe him?" (We've been talking about this debt and I know you've been worried about it and that you're looking forward to paying off your debt.
I can't imagine saying "Did you pay him money?"
If someone did something that was supposed to be at no charge (fix a part that broke while under warranty, for example) and I see my husband give him some cash, I might say "Hey, you weren't supposed to have to pay him. Did you just pay him?" - I wouldn't say "pay him money." (My husband would say "I just tipped the guy five bucks - he was on time and polite and seemed like a nice kid.")
Interested in Language