They can not use regular tribbles but if you have the one from the referral or the liberated-borg tribble you can use those.
I forgot my klingon is a liberated borg and they can use any tribbles so I made a fed klink and a regular klink; I logged in to test this and it works with both Fed Klinks and regular Klinks.
The Klink and the tribble still have the shock-reaction animation; however, one can receive the buffs from these tribbles.
The photonic-tribble buff is a nice buff for the klinks to have.
I am not sure how long it will last but right now Star Trek Online is 66% off and it is selling for $16.99 on steam. To get the photonic tribble, all one needs to do is refer and buy the game; a subscription is not required to get the tribble only the photonic BO and other perks require the subscription.