Back that thing up?

No, that's not how it's used.

You "back someone up" when you support what someone says. "I was late because there was terrible traffic. Jim can back me up on that."

You can also "back something up" by providing background facts. "The moon is made of green cheese." "Do you have anything to back up that claim?"

To say you accept what the other person says, informally, you can say "'l'll buy that" or "I can buy that."

Okay, if you've heard someone say "back that thing up," it's an idiom that means "Go back a couple steps in the conversation. Something you've just said doesn't make sense or something you've just said doesn't sound like it's true/accurate." You may also hear it as "Back up the truck a minute, there, buddy," or something similar. It's very informal.

If you want the "provide the backgroung facts" then you say "Do you have anything to back that up."

If you did hear it exactly as "back that thing up" perhaps you can provide a little more context if these explanations aren't making sense in your context either.