There are actual shotguns, which fall into the broad category of of weapons with the 'Blast' quality.
There are no rules for firing a slug from a Blast weapon.
You can certainly use a weapon with the Automatic property against a single target, but the Automatic rules are pretty specific. Doing this would just used your whole magazine to hit the target once.
There's no 3 round round option at all. Starfinder is pretty tight on the amount of attacks you can make in a turn, I'd expect a 3 round burst to merely increase your damage die by a step or something similar, rather than actually acting as if 3 rounds were fired and doing damage separately (or even combined.)
For bursts, the most properly balanced implementation would probably be to add the Boost property to some projectile weapons.
If you want shotguns that are their own whole subsystem, I believe Rogue Genius Games has a product for that (Starfarer's Armory: Shotguns), but you won't find it in 1st party core rules.
For other D20 systems, I think there are D20 Modern supplements with rules for the kind of thing you want, but I would be very careful about what rules you adopt. Because Starfinder uses increasing damage of single attacks as the primary scaling for higher level characters, anything that uses multiple attacks as a system has a large chance of breaking the game.