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
###Cait 25%25% faster AP regeneration if your HP is below 25% ###Codsworth +10
Codsworth
+10 Damage Resistance against robot's energy attacks ###Curie Heal
Curie
Heal 100 HP when your HP drops below 10% ###Deacon 20%
Deacon
20% more sneak attack and 20% increase in Stealth Boy duration ###John Hancock 25%
John Hancock
25% more Critical Hit if you have more than 250 RADs ###MacCready 20%
MacCready
20% greater chance to hit in VATS ###Nick Valentine 1
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%
Paladin Danse
20% more damage against ghouls, super mutants and synths ###Piper Double
Piper
Double XP for completing speech challenges and discovering new locations on the map ###Preston Garvey 20%
Preston Garvey
20% Damage Resistance and 20% more damage when outnumbered ###Strong 20%
Strong
20% more melee damage if your HP is below 25% ###X6-88 Increases
X6-88
Increases Energy Resistance by 20
Dogmeat
###Dogmeat NotNot 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.