I have found the baptism. He was considerably older.
From the parish records of St. Louis, Lotbiniere, Quebec:
(translated from the original French)
20 March 1860, baptized Joseph Eugene, born yesterday, of the legitimate marriage of Jean Baptiste Laliberte, farmer, and Anne Dunn, of this parish; sponsors are Leopold Lemay and Julie Auger.Father was able to sign his own name.
From the same parish records:
12 April 1847, after the publication of one bann of marriage, Jean Baptiste Boireault (?) dit Laliberte, farmer, of this parish, son of age of Jean Baptiste Boireault die Laliberte at Clotilde Chandonnais (?) of this parish, married Anne Dunn, daughter of age of Michel Dunn and Catherine Hargustine(?), of Quebec.
1861 Canadian Census
Lotbiniere, Canada East (later Quebec)
Laliberte, Jean B., farmer, aged 37, married
Dunn, Anne, aged 39, married
Laliberte, Alphonsine, aged 13
Laliberte, Antoinette, aged 12
Laliberte, Gustave, aged 10
Laliberte, Omer, aged 9
Laliberte, Georgina, aged 4
Laliberte, Eugene, aged 1
In the notes, they had lost two children in the past year, both of the flu, a boy aged 4 years, and a girl, aged 11 months.
From the parish records of Notre Dame, Quebec (City),
21 July 1820, Anne, born today, of the legitimate marriage of Michael Dunn, carpenter, and Catherine Harquatin(?).
There is an online family tree that indicates that Catherine's maiden name was Harquatinn. Note that in Quebec, women do not take their husband's names upon marriage (French civil law). In this same tree, Jean Baptiste's last name is Roireau dit Laliberte.
Jean-Baptiste Sr. and Clotilde were married 28 September 1812. He was the son of Joseph and Marguerite (Lemay). Jean-Baptiste Sr. was born in 1787 and did not die until 1881 and Clotilde after 1871, so unfortunately, there's another family folklore story that proves incorrect.
Lotbiniere is on the south short of the St. Lawrence River across from Quebec City.