No, but let's start with why you might think the contrary:
They are the same pixels. Therefore each pixel has the same noise-characteristics, the same dynamic range and sits behind the same anti-alias filter. Assuming your sensor does not out-resolve the lens, meaning your DX lens is good enough.
What you get is a 5 MP crop from a 12 megapixels sensor. That's a very nice 5 MP image but its still 5 MP, you can print a smallish print but it will degrade quickly as you get past an 8" x 10".
With 10 MP DSLR (APS-C or not), you have enough resolution to make a sharp 10" x 15" print. This is of course until noise-levels start eating away image details and lowering dynamic-range.
That is where the full-frame sensor's bigger pixel show there advantage, as you compare at higher ISOs, say 1600+ on modern DSLRs, then, your full-frame camera will pull-off the print much better. At some very high-ISO, I guess that you will get a better image form 5 MP full-frame DX-crop versus 10 MP APS-C but it's only going to be in such limited circumstances.