When Ezekiel was written
The book of Ezekiel consists of thirteen sections, each dated by the number of years since 'the exile of Jehoiachin' (597 BC), beginning in the fifth year. The total span of time for Ezekiel's recorded prophecies was about twenty-two years (c. 592-570).
Features of Psalm 137
This Psalm doesn't tell us exactly when it was written, and because it is poetic in nature, it could very well have been written many years after the events it describes (i.e. the author didn't need to necessarily be sitting 'by the waters of Babylon' at the moment he wrote the psalm). But it does provide several historical references that can help us to narrow down a time frame:
Babylon is still the dominant world power over the Jews: the Jews are in 'a foreign land', and the Babylonians are their 'captors'. This is the whole point of the psalm; the author is calling upon God to bring divine vengeance against the empire that has the Jews in exile.
Jerusalem / Zion has been conquered in some way, but we are not specifically told if this was simply the conquest of 597 BC, or the actual destruction of the city and temple in 587 BC.
Edom is specifically mentioned for its role in Jerusalem's defeat by Babylon (137.7). Edom was a frequent target of vengeful prophecies because Edom exploited Jerusalem's conquest by Babylon (Jeremiah 49.7-22; Lamentations 4.21-22; Ezekiel 25.12-14; 35; Joel 3.19; Obadiah; Malachi 1.2-5). However, there are earlier books that criticize Edom for apparently the same wrongdoing (Isaiah 34; Amos 1.11-12). According to the historical traditions in 2 Kings and 2 Chronicles, there were several occasions where Edom hassled the Jews before Babylon's conquests of Jerusalem. Although the two points above narrow down the time period, there isn't anything in Psalm 137 that explicitly places it after the 587 destruction of the temple.
Psalm 137 is certainly within the period of 597-538 BC, but it may specifically date between 597-587 BC due to the lack of references to the temple being destroyed. This would make it contemporary to the prophecies in the book of Ezekiel. So...
Yes, Psalm 137 was written for the same historical occasion as Ezekiel.