There actually is one good ST/TF crossover that I've read if you're looking for something already written. Not really along the lines of one going to visit the other though. It puts Ratchet and McCoy on the same ship. Can you just hear the snark? Let me dig up the link.
Matters of Medical Necessities (http://www.fanfiction.net/s/5907314/1/Matters_of_Medical_Necessities) Summary: Leonard McCoy thought he'd seen just about everything space had to offer in the years he'd been stationed on the USS Enterprise. But when Kirk orders an away mission that brings back a strange visitor with ties to 21st Century Earth, McCoy wastes no time in both picking his guest's brain…and pissing him off.
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timeasmymeasure
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Date: 2011-08-15 02:15 pm (UTC)Matters of Medical Necessities (http://www.fanfiction.net/s/5907314/1/Matters_of_Medical_Necessities)
Summary: Leonard McCoy thought he'd seen just about everything space had to offer in the years he'd been stationed on the USS Enterprise. But when Kirk orders an away mission that brings back a strange visitor with ties to 21st Century Earth, McCoy wastes no time in both picking his guest's brain…and pissing him off.
;_; I just wish they would update.