In D&D canon, half-goblins (but not half-hobgoblins) show up in the short story “Vision” (Roger E. Moore, in the Realms of Infamy anthology, 1994) and the video game Icewind Dale II (Black Isle Studios, 2002). As far as I can tell, they have never actually appeared in an RPG product or had RPG stats.
There are D&D 3e stats for both half-goblins and half-hobgoblins in Kingdoms of Kalamar (Kenzer and Co., 2001), but despite using the D&D logo, this is not an official D&D product.¹ There’s also Races of Ansalon (Sovereign Press, 2007) with “v.3.5 revised edition” stats for half-goblins in the Dragonlance campaign setting, but that doesn’t even use the D&D logo. And that’s the best I can find.
As for “why,” in Gygax’s conception of Dungeons & Dragons, the world was “human-centric.” Gygax considered this necessary to make the world intelligible for players—a world where humans were not dominant would, in his view, necessarily be very different from our own (human-dominant) world, to the point that it would be alien and confusing, and it would detract from the game too much to try to explain all of it to human players.
As a result, the demographics in early editions of D&D skewed heavily towards the human race, and it was stated explicitly that only humans could create “half-” races. This has not historically held absolutely true (particularly where dragons or fiends are involved), but even to 5e the overwhelming majority of hybrid D&D races are half-human.
And since half-whatevers were also half-humans, relatively few hybrids of “enemy” races show up, and those that do tend to have unpleasant implications, again particularly in older D&D. Half-orcs, obviously, exist, but much of their background remains highly problematic, even after considerable attempts to reclaim them. Newer human-with-some-X-traits races tend to involve more limited non-human ancestry, from further back in time, to avoid this (e.g. changelings, dragonborn, shifters, etc).
- Kenzer had sued Wizards of the Coast for copyright violations relating to the republishing of some D&D comics in a compilation CD, and part of the settlement deal included Kenzer and Co. getting the rights to use the D&D logo on their products during 3e.