Montiel has joined from Sevilla on a one-year loan deal with the option to make the move permanent next summer.
The 26-year-old is Nottingham Forest's fifth signing of the summer, joining Chris Wood, Antony Elanga, Matt Turner and Ola Aina in moving to the City Ground.
Montiel gained international recognition when he struck the decisive penalty as Argentina beat France to win last year's World Cup in Qatar.
The former River Plate right back also scored the winning penalty when Sevilla beat AS Roma in last season's Europa League final.
'I'm really happy to have joined this club,' he said.
'I'm coming here full of expectations, of getting to play and showing what I can do.
'I'm also hungry to win things, to win trophies, now's the time to show this.
'I've been watching the Premier League since I was child, I used to watch every game.
'As a player you want to play in the best league, and the Premier League is just that.'
Forest's chief football officer Ross Wilson added: "We are pleased to welcome Gonzalo and his family to Nottingham.
"He has already experienced some major achievements in his career and he is looking forward with anticipation to the next chapter with Nottingham Forest."
