I answered this assuming you were referring to Japan restricting their own citizens from leaving rather than the U.S. preventing Japanese migration from occuring

Ah, was asking about the opposite.

A quick google seems to hint at a quota of 4,500 immigrants restricted by the USA) from Japan per year but it doesn't directly source that number:

with the enactment of the Immigration and Nationality Act of 1965, which resulted in more than doubling Japanese immigration to the United States from less than 2,000 to about 4,500 per year

via

• http://immigrationtounitedstates.org/663-japanese-immigrants.html

see also:

• http://thehill.com/blogs/pundits-blog/immigration/259076-why-does-the-us-discriminate-against-immigrants-from-big

• http://www.tnvisabulletin.com/the-immigrant-numerical-contro/

• https://www.law.cornell.edu/uscode/text/8/1152

• http://immigrationroad.com/visa-bulletin/immigrant-visa-annual-limit-and-cap.php

• http://caracas.usembassy.gov/iv/categories-of-immigrants/immigrants-subject-to-numerical-limitations.html

My google-fu is weak, as I was not able to find much about modern Japanese immigration the US, nor a list of the per-country quota numbers.