Nope not on the first hit. The Red card will be used after the last hit
Source I just tested it on showdown.
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Cloyster used Icicle Spear!
The foe's Snorlax lost 8% of its health!
The foe's Snorlax lost 8% of its health!
The foe's Snorlax lost 8% of its health!
The foe's Snorlax lost 8% of its health!
The foe's Snorlax lost 8% of its health!
Hit 5 time(s)!
The foe's Snorlax held up its Red Card against Cloyster!
Alakazam was dragged out!
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//Jofly hope this helps