i had 5 HQ on elite beret while skilling up clothcraft, i was 1 under cap as my natural level, but i was using +3 items and +3 support. I leveled on them for 2 levels, but got all 5 HQ during the last level of skilling.
On my goldsmithing mule i've had unnatural luck with T0 HQs... including a plat. earring, many ingots and sheets, hairpins and rings. I think i've HQ'd something of every item i've skilled up with except for 3 items, and the mule is now lvl 65. All these HQs happened with support, however, and the support bringing the character over cap on the item. EXCEPT! The plat. earring i was still below the cap on the item, WITH support.