Williams spoke with the referee after hearing the alleged abuse when he was preparing to take a corner in the first half, and play was held momentarily as supporters were warned.
The Spain winger scored shortly afterwards and celebrated by tapping his arm, indicating the colour of his skin.
"I went to take the corner and I heard monkey noises," Williams told DAZN.
"There weren't many of them. There are stupid people everywhere... I hope this changes bit by bit."
Williams said his celebration was a response to those who insulted him.
"It was with a bit of anger, it's not normal to be insulted for the colour of your skin," he added.
Atletico Madrid midfielder Koke offered Williams "support and strength" after the game.
"In our society there is no place for people who insult like that," said Koke.
"Atletico Madrid and all our fans are in favour of Nico and against these situations, which should not happen in a football stadium or in society.
"It can't happen in the year we're in or the society we're in."
Spanish football has suffered a spate of racist incidents in recent years, many of which have been aimed at Real Madrid forward Vinicius Junior.
Athletic, fifth, are now six points behind fourth-place Atletico Madrid in the race for Champions League qualification with five league games remaining.
