I was there! Only 5 years old, but I remember it!
As I recall, one of two scenarios happened: either my dad was already watching it and I just so happened to walk in and start watching with him (which is how I first became a fan anyway) OR I was already watching TV, and either by flipping channels or random memory I found it on and insisted we watch it.
Either way, we had missed the whole thing except the end, with the three starships Enterprise trying to close the anti-time fissure. I remember as each Enterprise was destroyed, one by one, until Q said "Two down, one to go" which stuck in my memory when I rediscovered the finale some 4 or 5 years later on VHS. Either way, the last Enterprise went Kablooie! and I saw Picard sitting with his head n his hands in a room that I figured was the afterlife (I hadn't seen much of season two, and none of season one at that time, being born in 1989) so I thought at first Q was Picard's dad or something in heaven, chastising him for so carelessly throwing his life away. Again, I was 5 years old so I didn't "get it" but it was so cool when Q said "Well, if it puts your mind at ease, you've saved humanity once again!" At this point I must have wandered off again or been told to got to bed cause it was bedtime, because when Generations released later that year I was somewhat surprised. After all, wasn't the Enterprise destroyed in the anti-time rift? Was Q really so kind as to give Picard back his ship? Again, a five year old's mind doesn't work like an adults. Ah, but it was glorious.
A couple weeks later, I heard that TNG was over. My poor little heart was broken. I had just gotten used to watching Star Trek with my dad. And honestly, until Discovery came out, I didn't watch any new released Trek episodes with him again; the only shows we watched were TNG on VHS or DVD. DS9 and Voyager were unpopular in my household, and Enterprise had that theme song that I hated at first, so I missed the first few episodes, and by the time I started watching my dad had given up. I did start watching TOS when i was 7 or 8, but those reruns were shown on NBC at 4:30 PM so my dad wasn't off work yet. I will always be so grateful to the Kurtzman era of Trek: for the first time in over twenty years, me and dad have been able to tune in to watch weekly newly released episodes of Star Trek and watch it together.