Can players activate Quick-Play spells or Traps as face-down monsters are flipped face-up in battle?
It Depends On What the Card Does.

So, keep in mind that because the monster is being flipped face-up in battle, it means that we're in the damage step at the time of the flip. Because of this, there are some restrictions on what kinds of cards can be activated. A card/effect may only be activated during the damage step if:


It is a counter-trap card
Its activation is mandatory
It negates the activation of a card/effect (Note Activation, not Effect of)
It directly modifies the Attack/Defense of any monster(s)


Forbidden Chalice happens to hit one of the criteria here (modifies ATK/DEF), so it can be activated in the given scenario. If, on the other hand, one wanted to use Breakthrough Skill, one would not be able, since it does not meet any of the above criteria.