No, you cannot. Western Union and MoneyGram are two separate payment systems. They are not interoperable in the sense that you can send a payment from one system and pick it up on the other end.

Previously, in the world of exclusive arrangements, an establishment could only choose one, MoneyGram or Western Union. In many countries, this exclusivity has now been abolished (Spain comes to mind). So now, you can go to an Agent, that might be an agent of both Western Union and MoneyGram, in which case it just makes it convenient to pick up the transaction. But true interoperability doesn’t exist.

Its akin to asking if you can pick-up your Pizza Hut order from McDonalds? Won’t happen.