Toney shared images of the messages he received on Instagram, with the caption: "I wasn't even going to post this but I woke up angry."
Brentford said the abusive message was sent on Friday, after Toney scored twice in the match in West London.
"We condemn this discriminatory behaviour in the strongest possible terms. An attack on one of our players is an attack on all of us," Brentford said in a statement on Saturday.
"We expect strong support from the police, legal authorities and from Instagram's parent company, Meta, to ensure that the individual involved faces the full force of the law for this despicable hate crime."
Toney has been a key player for Brentford this season with eight goals and three assists in 11 games in all competitions for the West London club.