It's a bit hard to understand, I THINK that "that account" refers to a previous sentence. It's not a common phrase in modern English so probably not worth learning. I even googled it to make sure it wasn't just me, and got almost no results ;-) Please note that the sentence you posted looks like very old-fashioned English, I'm guessing it's from a book written over 200 years ago? While old writers like Shakespeare, Austen, Dickens, etc. are WONDERFUL they're not the best guides on useful modern English.

Merely on that account = not just because of that. She's trying to persuade her husband to visit the newcomer. ... are determined to go, not just for that, for they don't visit newcomers in general. Implying that paying a visit is important. Also see here: http://www.macmillandictionary.com/dictionary/british/on-this-that-account