Can I participate in the event on my side account(s)?

For the event, I'm afraid, no.
Since regular faerie quests are given out in Random Events, you may participate in those on your side account(s) if you happen to receive one, so long as you earned the NP for the requested item on your main account. (Remember, earning NP or items on side accounts isn't allowed.)

However, since the faerie quest event -- where you receive a "free" faerie quest every day -- gives out extra daily rewards, it is treated like any other event or daily and you may only participate on your main account.