I know Vulcan is 40 Eridani and Andoria is somewhere in Procyon. Anyone else?
Tellar Prime is supposed to be 61 Cygni.
40 Eri for Vulcan wasn't strictly canonical until
Strange New Worlds: "Charades" showed it on a star chart last week, although
Enterprise season 4 strongly implied it by putting Vulcan 16 light years from Earth. It was James Blish who first proposed 40 Eri as Vulcan's star in his adaptation of "Tomorrow is Yesterday" in
Star Trek 2 in 1968. I suspect the TOS writers originally intended Vulcan to be the hypothetical Solar planet of that name once believed to exist inside Mercury's orbit, and Blish knew that didn't exist, so he added dialogue to make it clear that "the Vulcan" (as he called the planet) was not
that Vulcan, but another one around 40 Eridani, which scientists at the time already considered a plausible candidate for hosting a habitable exoplanet. (TOS itself eventually clarified that Vulcan was not a Solar planet in "Amok Time.")
61 Cygni for Tellar and Procyon for Andoria are not canonical either, I believe, unless they've shown up in onscreen map graphics. In the
Star Fleet Technical Manual, Franz Joseph established the Federation's founding worlds as Earth, 40 Eridani, Alpha Centauri, 61 Cygni, and "Epsilon Indii" (typo for Epsilon Indi), adding two other nearby systems believed to be plausible candidates for habitable planets. (Joseph ignored that "And the Children Shall Lead" had established Epsilon Indi as the Triacus system.)
Joseph implied strongly that all of the founding worlds except 40 Eridani were human colonies, and did not explicitly identify 40 Eri as Vulcan. But the subsequent
Star Fleet Medical Reference Manual by Eileen Palestine, Geoffrey Mandel, and Doug Drexler identified 40 Eri with Vulcan, 61 Cygni with Tellar, and Epsilon Indi (spelled correctly) with Andor (as Andoria was then known). When
Star Trek Star Charts came out decades later, Mandel moved Andoria to Procyon, since ENT had established that Vulcan and Andoria were neighbors, and Epsilon Indi is on the other side of Earth from 40 Eri.
Almost all of the ones with Bayer designations (Greek letter followed by constellation name) really exist, although most are the wrong kind of star (usually too bright and short lived) to have habitable planets.
No, not really -- there have been a
lot of fake Bayer designations in Trek, going back nearly to the beginning with Delta Vega (mashing a Bayer-type Greek-letter prefix together with the given name of the star Vega, aka Alpha Lyrae). There have been a ton of others -- Alpha Onias, Beta Antares (another mashup with a real star name), Beta Niobe, Beta Cassius, Beta Renner, Beta Portolan, Gamma Arigulon, Gamma Hromi, Gamma Vertis, etc. And some that come close but aren't quite right, like Alpha Majoris (is that Alpha Canis Majoris or Alpha Ursae Majoris?) and Tau Cygna (should be Tau Cygni). Not to mention when they just mash two Greek letters together, like Omicron Delta, Delta Theta, or Omicron Theta, or use a single Greek letter as the name of the system, like Alpha II, Delta IV, Omega IV, etc. And Sigma Iotia, which looks like a misspelling of Sigma Iota. Sometimes they put the Greek letter at the end, like Ceti Alpha (it should be Alpha Ceti, a real star aka Menkar) or Maxia Zeta (just gibberish).
So if anything, I daresay the fake Bayer-esque names probably outnumber the real Bayer names like Alpha Centauri, Epsilon Indi, Omicron Ceti, Sigma Draconis, etc.
Then, of course, a lot of Trek systems use real, familiar star names like Rigel, Vega, Deneb, Antares, Altair, Regulus, Canopus, etc. Those are the ones that are almost always too hot and short-lived to have planets, since naturally it's the brightest stars that tend to be given names.