You might not think that these two setting would mix terribly well, but if you do, my friend, you’re just not boldly going quite boldly enough! Lemme show you:
Let’s start at the start: GURPS Cthulhupunk is set only a few years before the events of Star Trek: First Contact. It seems likely that the catastrophic war that was fought mostly to overthrow the corruptions of the mythos, which had crawled into every aspect of life – government, business, religion, you name it – by that point. But it clearly couldn’t have been entirely successful. There would still be pockets of Mythos-related activity on Earth – to say nothing of the vast numbers of them out there in space.
Moving ahead to the Enterprise era, if you don’t see the machinations of Yog-Sothoth behind the ‘Temporal Cold War’, you’re just not looking hard enough. Both sides are clearly being toyed with by some greater power.
In the Original Series era, there are a number of Mythos-like stories – “Who Mourns For Adonais?” has such elements, and the mysterious Mirror Universe may actually be a creation of Nyarlothotep. And the self-proclaimed “God” of the movie era may actually be Nodens.
The Next Generation/Deep Space Nine/Voyager era is perhaps the most obviously infiltrated by the Mythos. Of particular note are the shoggoth-run Dominion, the mysterious wormhole entities (who may also be related to Yog-Sothoth), the Silicon Entity, and of course, the Q Continuum – which, based on its primary representative, appears none-too-distinct from the Court of Azathoth.