The former defender, who wore Atletico Madrid's colours from 2010 to 2019, was speaking on Onda Cero's Radioestadio Noche ahead of Sunday's derby against Real Madrid at the Metropolitano.
Álvarez spent the summer trying to force a move to Barcelona, only for Atlético to refuse to sell, and he remains at the club. Godín, now 40, does not think the manner of that push has gone down well.
"Julián Álvarez is a really great player. He must be a good lad, but I think he made a mistake… The supporters' anger is normal. We all understand that he wants to leave, but it's how you leave, how you go about it.
"I think we all agree that he made a mistake. Now he'll have to take it on the chin and, if he wants to repair the relationship, make good gestures towards the fans," the Uruguayan said.
Godín compared the situation with that of Antoine Griezmann, a former team-mate.
"I don't know whether Julián Álvarez will repair the relationship. It cost Antoine a great deal and with Julián we'll see... Time will tell, but what is clear is that people are going to expect him, at the very least, to give everything on the pitch when it's his turn to play."
He was more upbeat when asked about Cristian "Cuti" Romero, the Argentina centre-back Atlético signed from Tottenham Hotspur last month.
"He's a player you see and think could fit in well at Atlético de Madrid. The authority, the calm with which he plays, the personality... It seems like he's been here for many years, I'm sure he's going to do well," Godín said.
Godín also looked back on the dressing room he knew, and on Giuliano Simeone, now a 23-year-old first-team player and Argentina international, who used to visit as a child while his father Diego Simeone was in charge.
"We used to make Giuliano Simeone cry in the dressing room and now he's a man," said the Uruguayan.
He added: "Giuliano used to come to the dressing room and Diego Costa would throw him in the jacuzzi, put him in the cold bath. A disaster... He'd go crying, 'Dad, dad, they hit me.' And the Cholo would say to him, 'but you like it.' And he'd come back, he was a kamikaze."
The first Madrid derby of the season arrives with Real Madrid second in LaLiga on 15 points and Atlético fourth on 13. Godín's message was straightforward.
"Sunday is a game to win, no matter what."