thought i might have remembered wrong so chatgpt'ed it and it largely concurred (i was on holiday for Roma so dont rememeber)...

Using a strict definition—Moyes answering questions from a group of journalists, rather than giving a one-to-one Everton TV interview—I found three qualifying matches.

Date Pre-season match Result Post-match press conference?
15 July Accrington Stanley vs Everton 1–1 No — Moyes was not presented to the media.
19 July Blackburn Rovers vs Everton 1–0 No — a media appearance was considered but did not happen.
22 July Everton vs Port Vale 2–1 No — played behind closed doors.
26 July Everton vs Bournemouth 0–3 Yes — Moyes spoke to the assembled media following the match. Post-match footage
30 July West Ham United vs Everton 2–1 Yes — post-match media questioning took place at Soldier Field. Contemporary report
3 August Manchester United vs Everton 2–2 Yes — contemporary coverage explicitly identifies a “post-match press conference following the United match.” beIN Sports report
9 August Everton vs Roma 0–1 No formal press conference found — Moyes gave post-match interview/comments, but the stadium’s press-conference suite was unavailable.

The evidence for no press conferences after Accrington and Blackburn comes from Everton correspondent Joe Thomas, who reported that Moyes was not put before the press at Accrington and that the anticipated Blackburn appearance did not happen. Royal Blue report

Therefore, the qualifying matches were:

Everton 0–3 Bournemouth
West Ham United 2–1 Everton
Manchester United 2–2 Everton

If your definition includes any recorded post-match manager interview, then Everton 0–1 Roma could also be included—but I would label it an interview, not a press conference.