The home side led 21-8 at the interval but became disjointed in the second half when all seven replacements had taken to the field for some much-needed match practice as Hill intends to field a strong side in the European Challenge Cup.
"Thankfully it appears that Matt hasn't suffered any break but he will be out for a few weeks," said Hill, the former Bath and England scrum-half.
"We played well for most of the game and looked well organised and in control. We are looking better and better at the moment.
"The main thing with this game was the psychological test where for the first time this season we were favourites though Gloucester were at the top of the table.
"I would have been grateful to come away with a straight win against Quins as it would have meant that we had come through the test. But to come away with a bonus point from our five tries so I am very pleased although we took the gloss off it in the last 20 minutes.
"We need to give our seven replacements some match practice but we didn't look quite as organised.
"It is now a question of the players getting used to being in the top three and playing with that level of expectation and being able to shoulder that pressure."
David Lemi scored two tries, while Bristol's other scores came from Shaun Perry, Alex Clarke and a penalty try. Ugo Monye, Andre Vos and Tani Fuga scored tried for the visitors.
Quins director of rugby Dean Richards remains confident that their luck will change when Stuart Abbott, Hal Luscombe, Andrew Mehrtens and Paul Volley are fit.
Richards said: "We didn't do ourselves any favours as we didn't turn up and we had one man in the bin and they scored two tries and Bristol didn't look back from that.
"They were more dominate than we were in the scrum area, our lineout didn't function and if we don't get our first phase right then we don't deserve to do anything especially coming to Bristol who have, I think, one of the most experienced packs in the Premiership."