I've looked into this a couple of times, and couldn't make up my mind. I think that probably means I should leave things as they are.
The article Ozzyjack links to is simply wrong when it says:
"Barbeque", in general, can be considered a misspelling. It doesn't exist in official publications and there is not specified anywhere that it is accepted as another form for "barbecue".
Many reputable dictionaries give it as a less common variant. As for "official publications", the website of the National Parks Service in the US uses both variants many times. But probably the writer meant some official guide to spelling - not realising that there is no such thing. Actually, where are they from, the writer(s) of this piece? Some place where people say "there is not specified anywhere that it is accepted".
It's certainly true that the
Q version is used less often, but it is used quite a lot. I wouldn't be surprised if there are people who haven't noticed that it can be written with a
C. Anyhow my new motto is, when in doubt, leave things alone. So I'm leaving
barbeque as a common word.