They’re headed to different directions.
Voyager 1 is going towards the constellation Ophiuchus. There’s several nearby stars in that constellation, including Barnard’s Star. However, the star it’s going to approach next is Gliese 445, which is currently in Camelopardis and about 17 ly away. However, that star is rapidly approaching the Sun and V1 will pass about 1.6 ly from Gliese 445 in 40,000 years. At that time, the star will be one of the closest stars to the sun at 3.45 ly and no longer in that constellation.
Voyager 2 is going towards the constellation Telescopium, which is in the southern hemisphere. Also in 40,000 years, it’ll pass by Ross 248 at a distance of 1.7 ly. At that time, Ross 248 will be the closest star to the sun. Currently, Ross 248 is about 10 ly away and in the constellation Andromeda, which is in the far north. V2 will later make a distant pass to Sirius (4.3 ly, about half the distance Sirius is to us now).
According to this chart Ross 248 will be closest to the Sun between 33,000 and 42,000 years. Gliese 445 will be closest from 42,000 to 50,000 years. Alpha Centauri will still be quite close during that period as well.
ETA: took a long time to research my answer, so it’s not too surprising someone ninja’d me.