Recently, when I was writing about dialogue tags, I had to look up whether the correct spelling is dialogue or simply dialog. I know I’ve seen both, and I found an interesting answer.
Dialog and Dialogue in British English
Dialog and Dialogue in American English
The shorter spelling started spiking in the early 1980s, especially in American English, and in many cases it’s used in a computing context—for example, when people are writing about a dialog box where you input text.
It seems that the computing use (dialog), again especially in American English, has started drifting over into the older “two people are talking” meaning.
Nevertheless, the AP Stylebook recommends the longer dialogue spelling. The Chicago Manual of Style doesn’t specifically recommend one spelling over the other, but it uses the longer spelling throughout its own book. My recommendation is to use the longer spelling when you are writing about conversations in a novel or play.