I confirm what lovlov says. I've tested on all 6 Shards girls (3 during teh 1st Shard Event and 3 during the 2nd one) - all 6 times girl cards traded for shards didn't count towards "Flirt with 2 R girls"/"Flirt with 1 SR girl"/"Flirt with 3 SR girls" missions. While getting them in any other wway (any kind of Gacha, any kind of Chance Time, as ranking/other rewards during Events or from the cases in Gambit / Coliseum Gacha boxes and even as gifts from Facebook campaigns/giveaways DOES count towards the Daily Flitring Missions. So i guess only Shards Trading Center isn't included in thes "flirting" mechanics.
Stopped playing all Nutaku's games (and, thus, using HBC and other related forums) around 06.2016