Rose said it was the first time he "sat back and took note", as the forward who left Liverpool for City in a £49m move in 2015 was jeered by Reds fans.
"It's probably the first time I have sat back and actually realised and took note of how much the Liverpool fans dislike him," Rose told BBC Radio 5 Live.
"I take that for granted whenever I'm watching Raheem. I think he just gets on with it and maybe it had a part to play in whatever happened."
Both players have since addressed the rest of the squad to apologise for the incident, and Rose said he could sense Sterling's apology "was from the heart".
"You could sense that and you could hear that," added Rose.
"Joe spoke afterwards and we were all proud of what they said. We are just happy that everyone has woken up in a better mood this morning."