You don't need lubridate for this. Are they integers or character? If they're integers:
df <- data.frame(YEAR=rep(1995:1999, each=12),
MONTH=rep(1:12,1999-1995+1))
df$new_month <- (df$YEAR - min(df$YEAR)) * 12 + df$MONTH
If they're character just wrap them in as.numeric()