Can my accountant or lawyer represent both me and my child in a family business sale?

For a lawyer, generally no. A sale between parent and child creates the same kind of conflict of interest as a sale between two strangers, since the seller and buyer have opposing interests around price, terms, and risk allocation, and professional conduct rules generally require each side to have its own independent lawyer rather than sharing one. This protects both of you: it ensures each side actually gets advice focused on their own interests, and it removes a common basis for a family sale to be challenged later as unfair or improperly advised.

An accountant's role is somewhat different, since accountants aren't bound by the same conflict rules as lawyers, but even there, independent tax advice for each side is often worthwhile given how differently a sale can affect the seller's and the buyer's tax position. Treating a family sale as informal enough to share one advisor is a common shortcut that tends to work against both sides if anything goes wrong later.