Not strictly speaking, no. You do need to sign in to a Google account to access the Play Store, which is synonymous with Android for most people, but it doesn't need to be a Google+ account.

Even given that though, the vast majority of Android apps don't require you to sign in with a Google account (or a Google+ account) and thus don't have access to the data associated with it. For a lock screen app to even require signing in is pretty strange.