Here's the Ballad Index entry . . . .

Marching Down to Old Quebec
DESCRIPTION: "We're marching down to (old Quebec/New Orleans), Where the drum is loudly beating, The 'Merican boys have won the day And the (British) are retreating." The soldier describes marching, and his plans to go home/to New Orleans/to visit a girl
AUTHOR: unknown
EARLIEST DATE: 1903 (Newell)
KEYWORDS: playparty soldier war battle floatingverses courting rejection Canada
HISTORICAL REFERENCES:
1775-1776: American attack on Canada. The chief battle of the campaign was fought outside Quebec on December 31, 1775
Jan 8, 1815 - Battle of New Orleans. Although a peace had already been signed, word had not yet reached Louisiana, which Pakenham sought to invade. Andrew Jackson's backwoodsmen easily repulse Pakenham
FOUND IN: US(NE,MW,So) Canada(Ont)
REFERENCES (2 citations):
Randolph 519, "We're Marching Down to Old Quebec" (2 texts, 1 tune)
Fowke/Mills/Blume, pp. 57-59, "Marching Down to Old Quebec" (1 text, 1 tune)
ST R519 (Full)
Roud #735
CROSS-REFERENCES:
cf. "Little Pink" (floating verses)
Notes: The history behind this song is somewhat confusing. Although America in its early years coveted Canada, and sent troops northward during the War of 1812, Americans never successfully attacked Quebec.
If the song refers to an actual event, it probably dates to the invasion of Canada by Montgomery and Benedict Arnold in 1775. Montgomery captured Montreal, and rendezvoused with Arnold to attack Quebec, but the assault of December 31, 1775 was repulsed. Montgomery was killed and hundreds of Americans killed, wounded, and captured.
As it turned out, both British and Americans sang about the invasion of Canada, with Americans lauding the capture of Montreal (which they obviously did not manage to retain) and the British celebrating the defense of Quebec.
This confusion may explain why the song was transferred to New Orleans, which was American property and where Jackson did repel a British army.
The Randolph version of this piece has more than a little connection with "Little Pink," and may even be the same song -- but at this point it's hard to tell; I've heard a "Little Pink" variant which goes in a completely different direction.
Roud tosses the whole family in with "Coffee Grows." - RBW