Is there way to see how many turret platings were taken after they fell?

There is currently no way to actively track this information via the in-game UI.

Since each plating destroyed grants your team 160 gold (as of writing this answer) this is definitely something to keep an eye on.

Luckily It's not exactly witchcraft to keep track of the platings manually. They plating always falls at the 14 minute mark so you can start checking yours and the enemies platings at around 13 minutes and you should be good.

If a turret is destroyed before 14 minutes just add another 800 gold to the team who managed to destroy it (if you know which champions, that's even better).

Consistently keeping track of gold income (not only platings) is a matter of practice but it'll improve your winrate once you've mastered it. Also:
Use the F-Keys in order to focus on your teammates!