In a word? No.
Star Trek never denies Relativity, and the resulting time dilation effects, but it never really deals with them. Even ships traveling at impulse (and hence, sub-light speeds) are never seen to have to deal with relativistic effects.
The five series and ten movies are incredibly inconsistent in their use of impulse power fractions and the time it actually takes to get somewhere. However, according to Memory Alpha's article on the subject, there's at least one reference out there that suggests that full impulse power is roughly equivalent to 0.25c. That velocity ought to be fast enough to involve a noticeable time dilation effect (0.10c is generally thought of as the threshold beyond which it's a real factor, although it's measurable with atomic clocks even today on the International Space Station!).
However, it's worth noting that time dilation only really matters if you have a ship that's traveling for a long period of time at a significant fraction of light, which most of the ships we see in Star Trek never do. For example, Julian Bashir suggested it would take "seventeen years, two months, and three days [...] give or take an hour." (DS9: A Time To Stand) for them to return to Federation space at sublight speeds when the warp drive on their Jem'Hadar ship is fried. That trip would undoubtedly have involved time dilation relative to their destination (Starbase 375), but we never find out because they never actually do it.
None of this means that time dilation is somehow not an issue in the Star Trek universe, so much as that it's never been an issue raised within the context of a story. The Stardate system was originally developed by Roddenberry and Sam Peeples (writer for "Where No Man Has Gone Before") to side-step the question of intragalactic time keeping. As originally envisioned, the only thing that mattered was that the stardates in a given script were internally consistent. As far as the show-runners of the Original Series were concerned, the stardate could in fact be different in different places, thus implicitly allowing for time dilation and relativity (and also the fact that they hadn't really decided how warp drive worked and how much it negated such things), while not really having to address it. (Memory Alpha: Stardate) Everyone just accepts that, as a starfaring culture, time is relative, and dates, even stardates, are conventions for counting local time.
The later series' passion for nailing things down to a real timeline relative to our current Earth calendar fudged this intention, but Star Trek retained the basic idea that space travelers just cope with time dilation if it happens, and never really address it otherwise.