Almost all of the reference books I consulted note that gov (or gov.) can be an abbreviation either for government or for governor. In the United States at least, that dual meaning can lead to problems of ambiguity: many U.S. readers are likely to interpret the Gov in a headline such as "Gov Relaxes Rules" as being short for Governor [of the state where the newspaper is published]. For that reason, in the United States, including the final t is a sensible practice.
None of the various sources I consulted recommend the abbreviation gov't; instead, they divide their preference between govt. (with a period) and govt (without one).
Proponents of govt. (with a period) include The Random House College Dictionary (1984), which actually prefers Govt. but includes govt. as an alternative; the Facts on File Concise Dictionary of Acronyms and Initialisms (1988); Encarta World English Dictionary (1999), Webster's II New College Dictionary (1999), The American Heritage Dictionary of the English Language, Fourth Edition (2000), and The New Oxford American Dictionary (2001).
Proponents of govt (without a period) include Merriam-Webster's Collegiate Dictionary, Eleventh Edition (2003) and The Concise Oxford English Dictionary, Tenth Edition (2002).
The modern legitimacy of both govt. and govt is well established by these sources. Nevertheless, my favorite bit of style advice on the subject is this from The Associated Press Stylebook and Libel Manual (1994):
government Always lowercase, never abbreviate: the federal government, the state government, the U.S. government.
The boxed message in your Ngram Viewer result is a warning from Google that its Ngram search tool didn't actually search for all of the terms you asked it to find—at least not in the precise form you specified when you entered them; hence, in this case, the results it provides in response to your search query exclude any instances of gov. or Gov. that may actually exist in the database of books it searches. I'm not sure what Ngram is tracking with the graph lines on the chart that it claims represent (Gov.) and (gov .), but you'll notice that its search results from Google Books (the links below the chart) include no results for either term.