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1.
One on one time with someone else. Usually to discuss life issues, and other important things that occupy the two conversing people's minds.

In the sense of spending time with your grandmother and asking for her assistance.
Tom: Hey do we need some Granny Time tonight?

Hariot: Yeah, we really do. I'm in a pickle and need to figure this shit out.
by grannywhite November 06, 2011
 
2.
The decision to reject the idea of a night out and instead stay in, have a nap, and drink a cup of tea.

One example is a Granny Nap, which is a small period of sleep during normal day hours, just like your nan. Both relaxing and refreshing and highly underrated.
Didn't go out last night, decided to Granny it up instead with some Granny Times.
by Rodgers25 April 12, 2009