Gareth Southgate's men won Thursday's World Cup qualifier 4-0 at a packed Puskas Arena in Budapest.
Hungary are to play two home matches behind closed doors, with a third game suspended, after being charged over racist and homophobic chanting at Euro 2020.
But as Thursday's match fell under FIFA's jurisdiction, a full house was allowed into a match marred by bad fan behaviour.
Deafening jeers welcomed England players taking the knee before kick-off and Sterling was pelted by missiles when celebrating the opening goal.
Further items were thrown onto the pitch, including a flare, and ITV reported that some monkey chants were aimed at Sterling and substitute Bellingham as he warmed up.
In 2019, England players were subjected to racist abuse in European Championship qualifiers in both Montenegro and Bulgaria.
England captain Harry Kane said he was unaware of any racist abuse but will speak to his team-mates.
"I didn't hear that," he told ITV. "I'll talk to the boys and see if they heard any of it.
"Look, we'll have to report it to UEFA as the rules with it. If it's the case then hopefully UEFA can come down strong."
England team-mate John Stones did not hear it either but he too hopes for action.
"I personally didn't hear it, I've been told about it after," he said.
"It is so sad to think that this happens at our games and I hope UEFA and whoever needs to take care of it does and I'm sure they will.
"We stand together as a team and we will continue fighting for what we believe in as a team and what we believe is right."
Stones added: "I think we definitely have the mindset of 'let the football do the talking' and tonight we did that.
"We celebrated our goals as if it is the last one we will score and that is an incredible thing to have as a team and as players.
"We live the game and are so passionate for the shirt and how we conduct ourselves on the pitch. It is our passion, let our football do the talking and that showed tonight."
Kick It Out chief executive Tony Burnett has urged FIFA to investigate the matter as a matter of urgency.
"Kick It Out was appalled to see the actions of some Hungary fans," he said. "We stand proudly in support of the England players and staff, who once again acted with strength, grace and integrity in the face of racism that has no place in our game or our society.
"This is not the first incident like this in Hungary, so we urge FIFA to investigate this matter urgently.
"Those responsible need to be held to account and steps need to be taken to ensure scenes like (this) are not repeated."
England sit top of their World Cup qualifying group with 12 points from four games.