If I'm not mistaken, we don't even hear about inertial dampeners in Star Trek until the advent of TNG. This doesn't mean that they haven't been retconned back in, for instance they are discussed in Enterprise despite that show occurring earlier in the Star Trek canon's timeline.
If Star Trek warp drives are true Alcubierre drives, then the ship itself does not actually move. Rather it plucks up a piece of "spacetime" and forces that forward in reference to regular spacetime, but stays motionless inside of it. Such a description is consistent with what we hear both in TOS and TNG, but the writers have never explicitly confirmed it (and they couldn't until recently, the Alcubierre drive was hypothesized in the 1980s and TOS predates it).
If warp drives are not, then it is a complete toss-up whether they are actually required. Considering Star Trek's propensity for technobabble, this could indeed happen at some point when a creatively bankrupt writer needs some melodrama.
My wild guess is that the writers will continue to make warp drives consistent (at least as well as they understand it) with the Alcubierre drive, and thus inertial negation is not required either before or during a jump to warp.