As the title says, will her Q proc them upon firing arrows? I've heard yes and no, so I wanted to clarify.
Based on the following information for the League wiki, Frost Shot will not proc Iceborn Gauntlets since it is a toggle ability. Iceborn Gauntlet shares the same passive effect as Sheen so it should function in the same way. You will need an actual spell cast such as Volley to get the effect rather than just toggling Frost Shot on or off. .
If you get the Iceborn Gauntlet effect to proc from a different spell cast, then it will apply to your next basic attack regardless if you have frost shot on. Here is an example of someone using Iceborn Gauntlet on Ashe by spamming volley to get the Iceborn Gauntlet effect:
A teamfight starts at 35:44 that shows the volley spam to proc Iceborn Gauntlet. You can see the same thing in a bit calmer setting when he is farming minions earlier.