No, the probe featured in Star Trek: The Motion Picture—known as V’Ger/Vejur—was not related to the Q Continuum in any fashion. Also, there was no mention of the probe travelling through time, only space. To be precise, it was said that the probe had travelled through a black hole and found itself on the “far side of the galaxy” where it crashed on a “machine planet” inhabited by sentient machines.
The machines believed the Voyager probe to be like them—much as a child—and built the vessel around it so that it might fulfill it’s primary programming: To learn all there is to learn and then return to its creator.
KIRK: V-G-E-R ...V-O-Y-A-G-E-R ...Voyager! ...Voyager VI?
DECKER: NASA. National Aeronautics and Space Administration. Jim, this was launched more than three hundred years ago.
KIRK: Voyager series, designed to collect data and transmit it back to Earth.
DECKER: Voyager VI …disappeared into what they used to call a black hole.
KIRK: It must have emerged sometime on the far side of the Galaxy and fell into the machine’s planet’s gravitational field.
SPOCK: The machine inhabitants found it to be one of their own kind, primitive yet kindred. They discovered its simple
twentieth-century programming. Collect all data possible.
DECKER: Learn all that is learnable. Return that information to its Creator.
SPOCK (OC): Precisely, Mister Decker, the machines interpreted it literally. They built this entire vessel so that Voyager could fulfill
KIRK: And on its journey back it amassed so much knowledge, it achieved consciousness itself. It became a living thing.
To date, no further information about the “machine planet” or its inhabitants has been revealed. However, Gene Roddenberry himself speculated that:
The “machine race” that built the probe around V'Ger was actually the Borg. However, he may have been joking at the time.