There are a couple of hints about it in both TOS and Voyager, and brick bats in Enterprise.
In TOS, the timeline has been muddled with by the Enterprise herself twice... once, in the Season 1 episode Tomorrow is Yesterday, and another in Season 2's Assignment: Earth. In both cases, they are interfering with the very 1960's they were filming in, showing it clearly to be an alternate earth. And that's ignoring The Guardian of Forever and that little drug-induced jaunt to the 1930's.
In Star Trek IV, we see Scotty give the molecular formula for Transparent Aluminum to an engineer in the 1980's. Further direct tampering.
In Voyager, we have *Future's End (3.04), which is interfering with the timeine.
And Enterprise has the whole Temporal Cold War... so lots of subtle changes happened.
So, from the very earliest days, Kirk, Spock, and McCoy have been altering the timelines.
It's even joked about in DS9 episode Trials and Tribble-ations that Kirk was a problem:
DULMUR: Be specific, Captain. Which Enterprise? There've been five.
SISKO: This was the first Enterprise. Constitution class.
DULMUR: His ship.
LUCSLY: James T Kirk.
SISKO: The one and only.
LUCSLY: Seventeen separate temporal violations. The biggest file on record.
DULMUR: The man was a menace.
It is further worth noting that the Assignment Earth backdrop is the 1960's, and they're launching a military payload on a Saturn V. This never happened in our timeline, so it was divergent by that point already. The Guardian of Forever episode's trip to 1938 or so leaves one man missing; phasered to death (by himself). By 1960, that may have had some effect. Also, we see interference in TNG with the time travel back to the 1880's. Plenty of divergences shown pre-1967. Even a woman running a major shelter and preaching is an oddity in the 1930's.
The Time's Arrow Episode of TNG is very much screwing with the timeline and with causality. It pushes back the divergence point into the 1850's. Because of interactions with Guinan, as well, we realize that Earth was being actively meddled with well prior to the TOS episodes' portrayals.