Lots of FTL technology uses a lot of power. To take your Star Trek example, IIRC they use antimatter reactors, which certainly is an advanced technology producing a lot of power, but not, I think, Kardashev scale II power, as they didn’t need particularly much antimatter, rarely really needed to refuel, and when they did refuel, they were more likely to be looking for fresh dilithium crystals™ than antimatter.
Moreover, Star Trek is a fairly good example of more advanced societies showing some restraint rather than trying to voraciously devour the universe following some bizarre hypercapitalist craze. Obsessed capitalists such as Harry Mudd tend to be identified as mentally unstable and somewhat hazardous, and treated accordingly. The invention of the Genesis Device is treated with great caution, and turns out to be more dangerous than good, even though it’s goal is to create Eden-like planets, not to generate uber-scale power generators. The needs of the Federation are amply served by powerful but limited-scale devices.
As you can see by the wiki for the Kardashev scale, level II gets into using planet-or-larger scale objects to generate massive amounts of power, which I can’t think of a Trek example of, though I haven’t watched much beyond the original Trek.
Doctor Who contains some examples of engineering on planetary and stellar scales, but in almost all cases, these tend to be pathological maniac projects which are highly immoral and require Time Lord intervention to prevent them from causing great grief to many.