As stated, does aircrack-ng when brute forcing a WPA2 handshake capture use GPU/CUDA resources such as a program like Hashcat does?
I don't think so. There's only one mention of OpenCL or CUDA in the source code, and it appears to be a leftover from code copied from John the Ripper.
Edited to answer your implicit question: if you want efficient WPA2 cracking on GPU, consider hashcat for the cracking, and hcxtools for conversion to the required format (hccapx). hashcat is superior to some other implementations because it has a rules engine that is implemented directly in-kernel on GPU.
As stated by @RoyceWilliams, it does not seem to use CUDA. For this purpose, you might want to look at Pyrit, since it leverages CPUs and GPUs on the cracking process.
NB: I wanted to leave this as a comment but I still don't have enough reputation.
Since I don't have enough "reputation" I cannot comment, but as Aero Wang's answer, you need to download cuda lib and recompile aircrack, I did this many years ago trying on a Nvidia Quattro, the aircrack-ng cuda runs very fast, if I remember well I could get 30000 passwords/sec instead of 5000/sec on a Xeon. But lately I try hashcat and it is better as built-in support for GPU is included, you don't need to recompile.