Your premise is faulty, because the
Enterprise was pacing the asteroid as it drifted through space. Asteroids don't move at anywhere even remotely close to relativistic velocity, even ones on interstellar trajectories. For instance, the recently discovered
interstellar asteroid 'Oumuamua had an estimated velocity of 26.33 km/s, which is less than a hundredth of a percent of the speed of light. Faster rogue stars and planets may get up to a few hundred km/s, still a fraction of a percent of
c. The time dilation at that speed would be insignificant.
Anyway, the numbers in "The Paradise Syndrome" don't add up. It took "several hours" to reach the intercept point at warp 9, and a bit over 59 days getting back at impulse. Now, onscreen warp travel has always been shown to be much faster than the supposed warp formulae, but if we go by the warp factor cubed model, then warp 9 is a minimum of 729
c, and if we assume "several" means, say, 5 hours, then we're talking a distance of about 3645 light-hours, i.e. 152 light-days, which is about 5 light-months. So to cover that distance in 2 months, they would've had to be going 2.5 times the speed of light, which of course is impossibly fast for an asteroid, let alone a starship at impulse speed.
More proof that it's generally best just to ignore the numbers in
Star Trek, because the people who come up with them rarely care if they make sense; they're just meant to give the audience a rough impression in passing, so they're unlikely to hold up to scrutiny.