What is the correct grammar for and barbecue?

"and" should be followed by a noun or noun phrase. "barbecue" can be used as a noun to refer to a meal or a grill. So "and barbecue" is grammatically correct when "barbecue" is used as a noun.

For example:

"We had hamburgers and barbecue for dinner."

"Can you grab the barbecue from the garage?"