There are at least two ways to interpret the question.
- Has the John Connor that we see in 2029 at the beginning of Terminator: Genisys experienced the events of Terminator 2 ?
- Does the future timeline shown in Genisys eventually evolve into the Terminator 2 timeline, or does it only connect to the timeline of the first Terminator film?
These are different questions, and I will try to answer them both.
Answer to "Question #1":
The events of Terminator 2 were almost certainly not experienced by the John Connor that we meet at the beginning of Genisys.
In Terminator 2, John and Sarah Connor destroy the Skynet project in its infancy and dispose of anything that can lead to its creation. This happened in 1995.
The John Connor at the beginning of Genisys is a product of the events of The Terminator — literally, Kyle Reese mating with Sarah Connor — and eventually becomes a commander in the war against the machines. The key piece of information is that, in this timeline, Skynet did become active in 1997. The date on which Judgement Day had occurred is confirmed at the very beginning of Genisys by Kyle Reese' voiceover.
If T2 had occurred in this timeline, meaning that the Skynet project had been destroyed in 1995, it is very unlikely that Cyberdyne or any other company or agency could have had Skynet fully developed and running in 1997.
Although T3 may no longer be canon necessarily, we see in that film that it took nine years for Skynet to become operational and self-aware after the setback of T2.
Note that the 2029 events at the beginning of Genisys do not require anything that John would have learned or experienced in T2. The privileged information he has is that:
- Kyle Reese is his father
- the machines will create a "temporal displacement device" as a failsafe; when the war begins to go badly for them in 2029, they will use it to send a Terminator back to 1984 to murder John's mother Sarah
- John will use the device to send Kyle Reese back to save her
These are things that he would have learned from Sarah from T1 alone. This knowledge is why at the start of Genisys,
in a final campaign to end the war, John leads a separate mission from the Colorado mission. The Colorado mission is intended to take out Skynet itself, while John's mission is to take out the temporal displacement device so that the machines cannot use it when it becomes clear that Skynet will not be able to repel the human forces in Colorado.
But despite their best efforts, events play out just as in the "history of the future" he has been taught, for John and his troops
find the temporal displacement device a few seconds too late, by which time the machines have sent the T-800 back in time to 1984.
At this point,
Kyle Reese volunteers to go back in time to stop the T-800. John picks Kyle over the other volunteers because he knows Kyle's destiny. (It is made absolutely clear later in the film that John knows that Kyle is his father and that he had deliberately withheld that information from Kyle in case events played out like this and someone had to be sent back in time. John wanted to ensure that Kyle would father Sarah's child, and so he withheld information from Kyle that might influence his decision-making and the subsequent course of events.)
Absolutely none of this requires anything that John Connor learned or experienced in T2. Combined with the fact that Judgement Day still occurred on-time in 1997, we can fairly safely assume that the John Connor of Genisys had experienced T1 but did not experience T2.
Answer to "Question #2":
Whether or not the events of Terminator 2 have occurred in John Connor's life by the start of Genisys, the events of Terminator 2 will not follow on from the 2029 events that we see.
In Genisys, shortly after
Kyle Reese steps into the machine to follow the T-800 back to 1984, a human soldier by the name of "Alex" grabs John's face. Kyle watches in horror from inside the temporal displacement device as John's face starts to come apart.
It turns out that
"Alex" is not a human at all, but rather Skynet itself in a humanoid form, known as the T-5000. As we see later in the film, the T-5000 infected John with a nanovirus — the T-3000 — which transforms John into a human-machine hybrid that has all of John's memories and skills but which is sympathetic to Skynet's cause.
So even if the machines were to have sent a T-1000 to 1995 to assassinate 10-year-old John Connor, thereby "initiating" the T2 timeline,
the T-1000 would not find John Connor in 1995, because Kyle Reese and Sarah Connor left 1984 together using a homemade version of the temporal displacement device, in order to prevent Skynet from proliferating as the Genisys operating system in 2017. If they were to have a child, it would be after 2017, and so John Connor would no longer be alive in 1995.
the future John's new loyalties mean he would not reprogram a "guardian" T-800 to follow the T-1000 back to 1995.
So, in other words, the timeline of T2 is completely eradicated because of
the T-5000's interference.
Interestingly, this interview reveals that
the T-5000 is not the Skynet that was destroyed in Colorado, but rather a Skynet from an entirely different timeline or universe that has developed the ability to jump between parallel universes. The appearance of this new Skynet is what prevents the usual T1 / T2 sequence of timelines from occurring as we know them.
Thanks to @Hypnosifl for the interview link!