Although he does not know her by name yet, it is Julia who is looking at Winston at lunch in Chapter 5.
Julia is sitting at the table to the left of Winston with a man who talks so incessantly that his mindless prattle is compared to the quacking of a duck. After a dramatic announcement from the Ministry of Plenty, Winston notices that
"...the girl at the next table (has) turned partly round and (is) looking at him...it (is) the girl with dark hair...she (is) looking at him in a sidelong way, but with curious intensity...the instant that she (catches) his eye she look(s) away again".
Winston is unnerved by the girl's observation of him. He wonders why she is watching him, and thinks that although she is most likely not a member of the Thought Police, she might be an "amateur spy", which would be even more dangerous. Winston "(does) not know how long she (has) been looking at him, but perhaps as long as five minutes". He is afraid that during that time he might have allowed his thoughts, which have not been orthodox, to have been visible on his face. Winston has noticed the girl sitting near him "for two days running", and is suspicious that she might be following him. When she turns her back on him, however, he wonders if it is all just a coincidence (Part I, Chapter 5).