The 24-year-old became the first African-born player to represent the Republic of Ireland when he came on as a late substitute during Tuesday night's 0-0 friendly draw in Hungary.
Before the game started, Ireland's players were booed by some Hungarian fans for taking a knee to support anti-racism. The same happened to England's players before their friendlies against Austria and Romania at Middlesbrough's Riverside last week.
Ogbene was proud of how his team responded, but now wants to see European football's governing body take action.
"I try not to worry about it," said Ogbene. "I just focus on ourselves. I feel like the group we have, it is diverse and everyone is together.
"We hope that UEFA will take stricter action and find a solution. It is a difficult task because it has been going on for many years. We won't find a result or solution overnight.
"I am so happy we continue to show how important it is to accept everyone for who they are and just educate people, and I am so proud of the team."
The England squad have already decided they will continue to take the knee during this summer's European Championship irrespective of the crowd reaction.
Ogbene was unhappy with the Hungarian reaction.
"I was obviously disappointed because we all go through different stories and different histories in our lives," he said.
"This is something that black people have been fighting for many years, discrimination and racism. There is no place in any sport, any place.
"I was quite disappointed. It is what it is. We stayed strong and I'm so happy that we as a team took the knee to show solidarity between us all."
The gesture of kneeling before games was popularised by American football player Colin Kaepernick in 2016 to protest against police brutality and racial inequality in the United States.
Sheffield United's Billy Sharp and David McGoldrick were hugely influential in bringing it to English football when the 2019-20 season restarted after the murder of George Floyd. It has been a staple of Premier League football since, though at Football League clubs some clubs have chosen to stop doing it.
Some individuals, such as Hull City's Josh Magennis and Malik Wilks, have decided to stop doing it, worried it might be losing its meaning, whilst others have criticised England's gesture as an act of support for the controversial political aims of the Black Lives Matters group. Manager Gareth Southgate has regularly made it plain his players are showing support for anti-racism, and not endorsing the politics.
Budapest is due to host three group matches at the European Championship, plus a last-16 tie. Hungary have not qualified for the tournament.
Seconds after coming on, Ogbene had the chance to mark his debut with a goal, but hit the side netting from a tight angle.
"Early on in the game, I was just hoping to come on and get the opportunity to score in front of their fans," he said.
"Maybe it is the best thing that it didn't happen because I might have let my emotions get the better of me with any sort of celebration I would have done."
Ogbene was born in Nigeria, but his family moved to Cork when he was seven.
He said: "To be titled the first African-born, I'm really blessed and it's a huge honour, something that I want to inspire everyone else to, to follow their dreams.
"I have got the first taste of it and I am hungry for more. I don't want to miss this opportunity and I will do everything I can to be an option for the manager."
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