Well, just in case anyone is curious, here is what happened when I put the hawks into the display. They went IMMEDIATELY to the powerhead that was closest to the overflow (the neon gobies spend most of their time behind the overflow box) and spent a lot of time looking behind the overflow box. (They didn't even explore the tank or "go to ground" as many fish do when in an unfamiliar tank, it was like they knew where the gobies were before they went into the tank.) The gobies were clearly nervous about the hawks, and within less than 5 minutes one of the hawks was swimming behind the overflow box and flushing the gobies out. I am still not sure the hawks are big enough (yet) to actually eat the gobies, but I had seen enough. I backed the hawks up as best I could and netted the gobies out and put them into another tank. (A few times during the process I chased a goby right past one of the perching hawks. Fortunately, they didn't swoop in a grab the goby. Maybe that confirms that the gobies are a little large, but it would only be a matter of time before the hawks grow, and now I can rest easy.)
Still curious if anyone has ever combined these species and what happened, but I don't plan to try it again.