Congrats!
Yes, the .45 will take the same backstraps as the full size .40. To be sure, I just took the backstrap off of my full size .40 and put it on my .45 and the backstrap from my .45 on my .40. They are the same.
However, I would point out that you might want to check out the stock .45 small backstrap that it comes stock with before buying others. The .45 is bigger in many dimensions. The .45 with a small backstrap feels about the same in the hand as a .40 with a large backstrap. While I personally like the small backstrap on the .45, I like the medium on the .40. My shooting partner uses the large backstrap on his full size .40 to make it feel like the .45 with a small. Unless you just like the hump that the medium has.