Speed, who captained the national side during his playing days, was the Wales coach at the time of his death in 2011.
Ahead of the semi-final with Portugal on Wednesday night, Swansea City's Williams said: "I was thinking about Speed on the plane here.
"I'm pretty sure he'd be proud of us. He's seen us grown up and now, with what we've achieved, he's always in our minds.
"When we achieve things we always think about him and hopefully he would be proud of us."