No, Star Trek doesn't exist in the Doctor Who universe

It's a bit silly to have to say this, but no, Star Trek and Doctor Who don't share a universe. No matter what some people will have you believe. Just because it's mentioned on air, doesn't make it canon. Waves were made in a trailer for the most recent season (called series in the UK) trailer saw the 15th Doctor (played by Ncuti Gatwa) proclaim that their universe and Star Wars universes were essentially linked.

In the first episode of their current season, "Space Babies" " the Doctor acknowledged that Star Trek, to some degree, exists in their universe when his new companion Ruby, alludes to the Tardis having teleportation powers like in Star Trek. To which, the Doctor replied, "We've got to meet them someday".

This of course got everyone on the internet excited about a crossover. A crossover that isn't likely to ever happen. See, despite some non-canon materials that see various versions of the titular Doctor character interacting with various Star Trek characters, those are all strictly non-canon interactions. They don't care and if we're being honest, they don't matter.

With the way the world of entertainment works, the issues with rights, streaming competition, and the like, the odds of Doctor Who and Star Trek actually existing in one another's universe is a near impossibility. Doctor Who is a British-owned property, owned outright by the BBC. Star Trek is owned by Paramount. Neither side is going to do business with the other, especially since the BBC has a working relationship with Disney.

The two sides will likely never do business together as independent agencies, so any true crossover won't ever happen. This means, that despite the number of times each show has referenced one another in media, they do not share a universe. It may bum some fans out but it's a simple statement of truth.

Every time a mention, be it minor or major happens towards the other show, it's not done with any more intention than to pop the fanbase a bit. They are throw-away lines meant for a minor chuckle, not a world-building plot you should invest any significant time.