Does having an extremely low intelligence (say, one or two) change the game's dialogue?

Word on the street is no.

Makes sense, both due to the more restrictive nature of the character's background (unless the USSA Military was that desperate, I don't think the mentally impared could get drafted) and for gameplay reasons (you can increase your SPECIAL stats at level up. The moron yesterday becomes the genius today sort of thing).

I disagree - The Pre-War scene could be straightforward but after you awaken from cyrosleep is when your SPECIAL points start to play a role.

I find it plausible that 200 years of cyrosleep could cause mental impairment which would fit for a low Int character

Could, but then it would not make sense to put points at the START of the game (before you cryosleep). just saying. I never redo my stuff when allowed.

But in the end, there was never much reason besides wanting to play a dumb (book-dumb anyways) character in any fallout games. At least fallout 4 allows you to get a gameplay benefit of low intelligence in the form of a perk in the luck tree.

New Vegas had some great low Intelligence dialogue moments. However, it was never consistent in its dialogue. One moment you are talking with an Intelligence of 1 and the next you are talking with an Intelligence of 5. The dialogue with your own brain was an absolute masterpiece, but the dialogue with the Chief Scientist at Helios One was good too. However, New Vegas had some of the devs that worked on Fallout 1 and 2 which was why we had the low Intelligence dialogue. Since Fallout 4 is made by Bethesda and Fallout 3 didn't have any low Intelligence dialogue, then there is none in Fallout 4.

TBH, I hardly put intelligence (or never) below four in NV due to skill points. Needed to max out those perk gateways.

You kind of want the title of your post to give us some information about the subject of said post.