Supporters have been angered by owner Assem Allam's plans to rename the club, although his proposals were dismissed by the Football Association in April.
Allam has since put the club up for sale, but some of the fans have continued to voice their disapproval of the situation.
"My appeal to all of the fans is that they can have their thoughts but once it gets to matchday, we need them to get behind the team," Sky Sports News quotes Bruce as saying. "We need them to do that because if it becomes fractious then it becomes difficult. When it becomes difficult then it can affect the team. That is why I appeal to the fans.
"We don't want any protests - even though I know nothing's changed on their behalf with the name-change issue. We want the supporters to get behind the team, that's my concern, and I'm sure they will.
"The protests hit their peak against Crystal Palace last season, and it seemed to wane after that. We lost that game and we should have won if I remember. We need all the supporters, if they can, to support the team. Of course they can keep their opinions - but not on a match day because that just harms everybody."
Hull are currently 12th in the Premier League table.