Roddenberry and Paramount haggled quite a bit over whether the pilot episode would be a standard hour, 90 minutes, two hours, etc. The original story just concerned Farpoint station and the jellyfish aliens. When it was finally agreed that it would be a 2-hour pilot, the Q story was hastily concocted to fill out the extra time. If I'm not mistaken, I think D.C. Fontana came up with the Q parts.
No, Fontana wrote the 90-minute draft dealing with Farpoint Station and the star-jellies, and then Roddenberry tacked on the Q subplot when an extra half-hour was added. The Q material is pretty clearly Roddenberry's work, since it's ultra-preachy about humanity transcending its past, and rehashes a lot of familiar Roddenberrian tropes like the superbeing testing humanity and the near-future WWIII that humanity will have to endure before coming to its senses. Indeed, the portrayal of "the Post-Atomic Horror" and the post-WWIII history feels very much like a reworking of ideas from Roddenberry's Genesis II/Planet Earth
Even still, you can tell that the episode is heavily padded to fill out the two hours. The pacing and the editing are kept deliberately slow so as to run out the clock. It's essentially a one-hour story being told in two hours, and the episode is hurt by that.
Well, it's a 90-minute story being told in two hours. Given all the characters and ideas they had to introduce, I'm not sure they could've fit it into one hour (or rather, 42 minutes).
I think the same thing may have happened with the pilot of Paramount's War of the Worlds: The Series
a year later. That pilot periodically veers off into a subplot about the disintegration of the lead character's relationship with his girlfriend (Gwynyth Walsh) due to his obsession with his work, and in the second half of the pilot that subplot is just abandoned and never mentioned again. It's completely tacked on and unconnected to the rest of the story, and seems to take up about 1/4 of the runtime, so I suspect it was the same situation as the "Farpoint" pilot, a subplot added to flesh out a 90-minute story to 2 hours. At least the Q subplot was tied into the main story of "Farpoint," however awkwardly.