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Say Sparta controls Byzantium and starts the winter move with a fleet in the Hellespont.
Can Sparta quarter that fleet by moving into 1) the unnocupied Sea of Marnara and 2) into the port of Byzantium?
The Hellespont is not adjacent to the city-state of Byzantium so Sparta cannot make a free Quartering Move to Byzantium. Byzantium is not a Home city to any Spartan unit so he cannot make a free Wintering Move there.
He could, however, make a group move--which would cost 1 action; or maintain the fleet at the cost of 1 action.
The question is whether the port is part of the sea area or the land area. If the port is considered to be part of the land area, moving from Hellespont to Byzantium would be 2 moves (so no free quartering move):
First move: Hellespont => Marmara
Second move: Marmara => Byzantium
The port is part of the land area. Your movement analysis is correct: Hellespont -> Marmara (1 move); Marmara -> Byzantium (1 move). Total of two moves.
But I think we need an official answer here.
This is official.