Do followers in Fallout 4 have any perks?

No, companions themselves do not have perks that you can select but they will grant additional perks if you make them happy enough. These perks are as follows:

Cait

25% faster AP regeneration if your HP is below 25%

Codsworth

+10 Damage Resistance against robot's energy attacks

Curie

Heal 100 HP when your HP drops below 10%

Deacon

20% more sneak attack and 20% increase in Stealth Boy duration

John Hancock

25% more Critical Hit if you have more than 250 RADs

MacCready

20% greater chance to hit in VATS

Nick Valentine

1 more guess when hacking, additionally, the wait time if you fail a hack is decreased by 50% if Nick is your current companion at the time

Paladin Danse

20% more damage against ghouls, super mutants and synths

Piper

Double XP for completing speech challenges and discovering new locations on the map

Preston Garvey

20% Damage Resistance and 20% more damage when outnumbered

Strong

20% more melee damage if your HP is below 25%

X6-88

Increases Energy Resistance by 20

Dogmeat

Not talking about actual perks in the game but as per the comments there are some unofficial perks relating to Dogmeat, in that he doesn't count as a companion in terms of Lone Wanderer, doesn't talk to you, and doesn't judge your actions. For all intents and purpose, having Dogmeat with you is like not having a companion with you while having a companion with you.