Oh, you're quite right. 'Axed' is in Anne of Green Gables, ins't it?
Oh, this is a classic.
Looking over the the OED entry, it seems that either 'to ax' is the original, or that 'to ask' and 'to ax' have been two equivalent versions from Middle English onwards.
c. 1000, we have both 'acsian', which looks to me like the forerunner of 'ax', and 'Ic ahsize', which looks like a forerunner of 'ask'
Chaucer has 'I axe'
A sermon of Archbishop Latimer's is recorded as using 'axe', while at much the same time, Shakespeare is using 'ask'.
The spelling to 'axe' seems to be coming up pretty evenly through the 16th centuries, in equally high status texts.
After that, the OED appears to be recording 'axe' as the dialect version and 'ask' as the RP.
May be one of those situations where the US / Canada has kept one 16th century version, perhaps dependent on the immigrants' regional dialect, and the UK RP has moved to a different 16th century version.