Tolkien's elves have better communication with animals than humans (like riding horses without needing a bridle or other control device) but is there any indication they keep pets?
The closest I can think of would be Huan, who was kept by Celegorm as a gift from Oromë, and then later bonded with Lúthien:
[T]he chief of the wolf hounds that followed Celegorm was named Huan. He was not born in Middle-earth, but came from the Blessed Realm; for Oromë had given him to Celegorm long ago in Valinor, and there he had followed the horn of his master, before evil came.
The Silmarillion III Quenta Silmarillion Chapter 19: "Of Beren and Lúthien"
I include Huan here mainly because he's the only animal I can think of who formed an emotional connection to an Elf, which is to my mind the thing that distinguishes a "pet" from a "kept animal."
Other than this admittedly shaky case, no; there is no evidence that the Elves kept domesticated animal for purely emotional reasons. Even Huan was principally a hunting hound, and would presumably have had a very different relationship to Celegorm than I have to my own dog.
Sometimes in the Silmarillion nightingales and other birds are mentioned around Melian, the Maia. Even in Menegroth were there birds, so they must live among the elves.