We know Goku in Super Saiyan God and in Super Saiyan Blue uses God ki. And apparently, he can also use God ki in base form. So, my question is, is Goku using God ki when he uses Ultra Instinct? Does the anime show something about this?
Goku is not using God Ki while using Ultra Instinct. We know this definitely based on Episode 110, 116 and 129. In Episode 110, When Goku used Ultra Instinct Omen in his fight against Jiren, Roshi and Tien were able to sense Goku's energy. Even Piccolo was able to sense his energy as he made a statement that he was becoming stronger during the course of battle.
Also, during Goku's fight with Kefla, we see Kefla being able to sense Goku power up increase his energy and hence responded by doing the same and even Jiren was able to sense this energy and wake up from meditation during their course of battle.
The "heat" which has been mentioned during all the times Goku turned Ultra Instinct , is a rise in Energy. We know this definitely in Episode 129 after Whis talks about the same. Also, When Goku pushes back Jiren's barrage of attacks, Jiren mentioned that Goku had generated this tremendous amount of heat to land a decisive blow on him and defeat him for which he must respond.(This was when Goku was powering up). When Jiren powered up before launching his attack on Goku, Toppo remarked that Jiren was "Getting way too heated up(Generating a lot of heat)", which in other words is a lot of energy.
We know the Super Saiyan God transformations are all about Ki control and Energy levels can't be sensed. Hence the above mentioned occurrences confirm without a doubt that Goku is not utilising God Ki during the Ultra Instinct Transformations
Yes Ultra Instinct needs God ki. In chapter 66 of the manga, Vegeta and the rest of the z fighters were giving Goku there energy but it wasn't enough. It was stated that for Ultra Instinct godly ki is needed. That is why once goku got Uub's energy he was able to go Ultra Instinct and destroy Moro's gem.