Is there a way to check whether I unlocked the Destiny nameplate?

I'm fairly sure the only way to know is to log into the game. If you got the nameplate it will either be in your inventory or at the Special Order vendor.

As mentioned by Eben, the nameplate should initially be in your inventory, or delivered via a special order, when you first log into the game. It is also possible, however, that you unlocked the emblem at an earlier date, and since dropped or removed it from your inventory.

With the release of The Taken King, you will find access to an Emblem Collection, where you will be able to retrieve any unlocked emblems. This includes the Sign of the Elders emblem.