Can Dianas energy be used against an opponent who is in an open showdown (for example to play gust when they attempt to conquer an empty battlefield?)

Yes — you can play Gust during an opponent's conquest attempt, and Diana's trigger works there too, but Diana doesn't produce energy herself.

First, a clarification: Diana, Lunari's ability is a trigger ("When a showdown begins here, you may pay [1] to Predict, then reveal… draw it if it's a spell") — it spends [1], it doesn't generate it. That trigger does fire when the opponent starts a non-combat showdown at her battlefield on their turn (FAQ #9373), and you can pay the [1] then.

The energy comes from your runes, not Diana. Your rune pool empties at the start of each player's Main Phase and end of each turn (Rule 167), so by the opponent's showdown it's empty — but basic runes have Reaction Add abilities ([E]: [Reaction] — Add [1], Rule 164.2.a), and Reaction works on any player's turn in any state, so you just exhaust ready runes to pay Diana's cost and/or Gust's [1] cost.

Gust is a Reaction spell, so it's legally timed during the Showdown Open state of that conquest attempt (Rule 813). If it bounces the moved unit (3 Might or less), the showdown still continues until all players pass Focus, but with no unit remaining the attacker can't establish control when it closes — so no Conquer happens (Rule 348.2.a).