Yes, at least once.
Barton is referred to as Hawkeye in Age of Ultron:
Laura Barton: What about Nat and Dr. Banner? How long has that been going on?
Clint Barton: Has what? [Laura laughs]
Laura Barton: You are so cute.
Clint Barton: Nat and...and Banner?
Laura Barton: I'll explain when you're older, Hawkeye.
Romanoff is referred to as Widow in Age of Ultron:
James Rhodes: Come on, Cap.
Steve starts pulling on the hammer and manages to budge it a little; Thor looks a little alarmed. Steve still fails to lift it; Thor laughs with relief]
Tony Stark: And?
Bruce Banner: Widow?
Natasha Romanoff: Oh, no no. That's not a question I need answered.
Tony Stark: All deference to the man who wouldn't be king, but it's rigged.
Clint Barton: You bet your ass.
In Avengers, the Russian general calls Natasha "The famous Black Widow" in Russian
Specifically, he speaks the line “Знаменитая Чёрная Вдова” (“Znamenitaya Chyornaya Vdova”), subtitled onscreen as “The famous Black Widow”; Google Translate confirms the translation.
A computer screen also has their names as:"Natasha Romanoff. Black Widow" and "Clint Barton. Hawkeye"
At 03:04 in this YouTube clip:
Credit goes to @ash108 for finding this.
During the Battle of New York, Natasha calls out to Barton using his Hawkeye moniker
See the very beginning of this clip
Credit goes to @UselessInfoMine for finding this.