Fans have tried to prove it using a book titled "Red Dead" by a "J. Marston", but that is only one of many easter eggs scattered all over GTA 5 about Red Dead Redemption. Using easter eggs as prove of something is a bit pointless since we are talking about easter eggs, which are small nods to pop-culture/other titles added by the devs for us to find, GTA is full of those.

However, I am a 100% confident that both games aren't connected, because both projects differ a lot in tone, world, and lore.

The best proof that I've seen that the worlds are connected is probably from Francis Sinclair. There are a lot of hints to suggest that he is or once was a high-ranked member of the Epsilon Program which any GTA V fan will know is a cult active in Los Santos.

Francis Sinclair doesn't actually play a role in the GTA HD Universe, he's just a reference to Epsilon, but what he discovers in Red Dead is a mural depicting historical events that have already occurred in GTA 5, nothing GTA related. The character's sole purpose is to show us more about what we don't get to see after 1914. It's a cool nod to GTA 5 nonetheless.

Well, there is Sinclair and epsilon, the book by who's assumed to be jack. Roads and areas in GTA v named after locations in red dead, the state of New Austin is confirmed to exist in GTA online. And the president from red dead 2 has a portrait in GTA V aswell. I don't think it's as far fetched as some of you make it out to be. Alot of rockstars series take place in the same universe. GTA 3D Universe and the GTA HD universe both are part of the manhunt and bully universe. I figured it's mostly agreed that red dead and gta are in the same universe. There isn't any reason for it not to be. It's not like it's a Fallout and Elder scrolls in the same universe theory.

But then again, the state of New Austin has never actually been made canon in the GTA HD Universe, and it wasn't even mentioned by name.

On June 7th, 2016 a new buyer for the player's warehouse was added in the Further Adventures in Finance and Felony update in Grand Theft Auto Online. The company's name, New Austin Capital, is derived from the state of New Austin; however the use of the name doesn't mean that it's based in New Austin itself, the name is simply derived from it.

This is what we can know for sure. In addition to that, in that same year the teaser for RDR2 came out, so it was most likely a hint rather than a proper connection since it was only done once years ago.