We might talk about the Presidency of Obama, to mean the time he was in office. As in, “during the presidency of Obama, Don’t Ask, Don’t Tell was repealed”.
What is the equivalent word to refer to a governor’s time in office? Governorship?
Yes, governorship.
One meaning of the word “government” is the tenure of the office of governor, so in that sense “government” is to “governor” as “presidency” is to “president”.
“Governance” would be another possibility but, again, the word has other, more commonly-used meanings.
So “governorship” tends to be used because it has fewer alternative meanings. But none of the three terms would be incorrect.
And of course for both Presidents and Governors you can talk about the “administration”. (Under the Obama administration, such-and-such happened; during the Long administration so-and-so was charged with corruption.)
Governorship is the term that is used. The dictionary definition of the word is “the duties, term in office, etc., of a governor.”
Government does not have that definition in any dictionary that I checked. Well, my old Webster’s does say “The term during which a governing official holds office” but labels it obsolete.
Gubertime, obviously. 
The Oxford English Dictionary has this definition of government: The position of governor; tenure of this.
The usage may be obsolete in the US, but it’s quite common in Australia. (Perhaps this is a revival?) Political campaigning and political discourse has been quite americanized in the last couple of decades and (despite the fact that we don’t have a presidential system) successive administrations are referred to (and refer to themselves) by the name of the Prime Minister. “There were three deputy prime ministers during the Howard government.” “Economic factors at play during the Hawke government included globalisation and industrial relations reform.” And so forth. In these examples, “the X government” refers to the period of time during which X presided over the government of the country. In the Australian context, X’s title obviously wasn’t “governor”, but the term would work just as well if it were.
Could be. But it’s never used in the U.S. and your usage sounds strange (and somewhat “British”) to my ears.