Tell that to the fans who think that ENT only exists because of a changed timeline in First Contact.
Disproven in "Regeneration" (ENT), "These Are the Voyages..." (ENT),
Into Darkness, and
Beyond (as in each single episode alone makes it impossible). Why are we still talking about this?
Does it? There's nothing in Kelvin Trek to suggest that the Enterprise we see there is the first of the name, right?
The non-canon
Countdown to Darkness tie-in comic did have a pre-1701
Enterprise, however since it's not canon (and the comics are riddles with continuity errors), the answer is no.
Star Trek isn't consistent on this front - Assignment Earth's plot, for example, rests on the reveal that the Enterprise's intervention was part of history already. STFC showing the assimilation of Earth scuppers the idea that the E's trip back in this case was a predestination paradox, so we must conclude the timeline was altered by the Borg, and then altered again by Picard and co. The timeline they created is at least to a certain extent, alternate. Zephram Cochrane and Lily had knowledge of the distant future, and several people die who didn't the 'first time round'.
There's no inconsistency with some time travel events being predestination paradoxes and others being accidents. As far as the FC incident, who's to say that the multiple changes in the timeline were not part of the proper flow of events (in other words, that the Borg Earth timeline briefly existing before the crew of the E set things back on course was "supposed" to happen). Also, FC was a predestination, then Cochrane and Slone "aways" learned about the future and all that.
(If nothing else, FC implies predestination with Riker given Cochrane one of his quotes early on and is confirmed to be predestination in "Relativity" [VOY] and "Regeneration" [ENT]. If the only hassle is seeing the Borg timeline before )
I find that hard to believe, considering the only time it ever actually HAPPENED on Star Trek was "Time's Arrow." In other works of fiction it's actually something of a mind-bender, and it's incredibly hard to convincingly write stories around.
Going off the top of my head, "Assignment: Earth" (TOS),
First Contact, "Future's End, Parts I and II" (VOY) dealt with them. It was even considered to make
Voyage Home a predestination tale at one time.
I don't even believe in Predestination Paradoxes. To me, it just means there is a previous timeline that set up events for a loop and we are not privy to what may have happened in a previous iteration / timeline / due to outside 4D manipulation.
"Trials and Tribble-lations" (DS9) establishes that they exist in the
Star Trek universe (and co-exist with non-predestined time travel).