The first part of your question, is Einstein's special theory of relativity mentioned in the original series, can fairly straightforwardly be answered in the negative. Searching online script databases for "relativity," "relativistic," "dilation," "contraction," "Lorentz," and "transformation" yield no relevant hits. There appears to be only a single mention of Einstein and that only as an example of a great scientist without specific mention of his work.
(The same applies to the pre-TNG movies; the only mention of Einstein is in ST:TWoK in a list of scientists.)
The second part of your question suggests a bit of confusion about what the special theory of relativity is. "Special" relativity is special because it is a specific subcase of General Relativity, to whit the subcase where the curvature of spacetime can be ignored. That means that analysis of time-bending effects of gravity wells, like black holes, properly belongs to General Relativity, not Special Relativity (which is basically the Lorentz transformations plus mass-energy equivalence).
If we broaden the scope of the question to match its apparent intent — mentions of General Relativity — the first explicit mention in a TV show or movie appears to be in the Star Trek: The Next Generation season 3 episode "The Price":
DATA'S COM VOICE: Captain, we have penetrated the outer event horizon of the wormhole. Sensors read severe doppler shifts...
(distorted)
What the bridge crew is hearing is gibberish. Picard frowns.
WESLEY: It's the relativistic time dilation, sir. Time is moving more slowly inside their part of the wormhole. I can process the audio to compensate...
