Yes, the custom of counting people as one year old when they're born is called kazoé-doshi in Japanese, but I've seen the word only in older literature or historical documents.

One custom that is still alive, though, is of counting years by imperial eras instead of by the Western calendar. On an official form, you're just as likely to see Heisei 22 as 2010, and even in conversation, people will talk about having been born in Showa 35 (1960) or Heisei 2 (1990).