No.
Spires Fire during the
Spires Fire step of a player's
Turn, targeting the
Active player's units. A
Landmark Minion is not under the
Control of any player so spires will never
Attack them. If a
Landmark Minion is under the
Control of a player, it can be attacked by that player's spires because it is controlled by the
Active player when the
Spire attacks.