Well, you can.
There's a system a mere 50 lyr away from starting system Dromi, and only 34 away from Sol known as Wolf 636.
There you can find 3 locations that can and will spawn Thargoids.
Two Distress Calls, and the Riker Class Prison ship in the system contain extra terrestrial encounters: at the Distress Calls, you have a 40% chance to find Interceptors. Could be a Cyclops. Could be two Hydra.
Meanwhile at the Prison ship, you'll find a mission you can't complete, and some Thargoids will show up. Could be scouts. Could be a Clops. Could be a 2 Hydra. Could get several waves of Xenos.
As a warning, since you'll likely see the real and quite hostile nature of Thargoids at the Prison ship, you might want to pick the Distress Call if you aren't looking to fight them.