Regan went off with a shoulder problem and his replacement David Blaney was also stretchered off in the final minutes with a suspected broken ankle.
England hopeful Dan Ward-Smith, who missed the chance of a place in last season's RBS Six Nations and the World Cup due to a dislocated knee, could again miss out on international recognition in the new year due to a suspected broken hand.
"He has broken his hand and gone for a scan. If it is clean break, he is likely to be out for six weeks but if it's splintered, it could be eight weeks. It's very unlucky for him. Last year, before he was badly injured, he was out for six weeks with a thumb problem," said Hill.
All the points came in the second half with Bristol scoring tries through wing David Lemi, Blaney and replacement prop Alex Clarke.
Fly-half David Hill kicked one penalty with his replacement Jason Strange adding one conversion.
Quins' points came through a well-taken try from full-back Tom Williams after a kick ahead from acting scrum half Simon Keogh, while Danny Care was in the sin-bin. Fly-half Adrian Jarvis landed the extras.
Hill added: "We should have kicked on and got the bonus point but if I was told we would have three wins out of four at the start of the competition, we would be delighted to be equal top.
"Having said that it was points dropped, although we are not a team that scores many bonus points as our game has been built on set-piece and defence and the attacking game will, I hope, develop over the next couple of years.
"With the possession and the chances we had, to come away without a bonus point has taken the edge off the win a bit."
Despite being content overall Hill went on to question some of his team's tactical play.
He said: "Quins made us work very hard, they defended well and were very enthusiastic. As it had been dry we started to ping the ball around unnecessarily. Rugby hasn't changed for hundreds of years. You have to go forward, suck in the defence before you move the ball wide.
"Even though it wasn't in our game plan, we moved the ball wide against four defenders and that isn't our game.
"There'll be a big crowd here at the Memorial Stadium for the visit of Cardiff Blues next month and I think it will go down to bonus points to decide who wins this pool."