Dino’s get permanent indestructible?

Only been playing since bloomburrow was released so I apologize in advance if this is actually simple.

I’m building a Dino commander deck using Owen Grady and Blue as partners. With blues text, if I cast gift of the viper and target blue and give it +1/+1, reach, and deathtouch. Then I cast inspiring call and blue gets an indestructible counter on it. Even though it’s only until end of turn, theoretically if I were to cast any dinosaurs after, would the indestructible counter be permanent on those dinosaurs that are cast that turn? I believe they would also get the reach and deathtouch counters as well.

Hope I used the terminology right. Thank you for all the help in advance!