(covers information from several alternate timelines)
Quantum matrix is a term referring to a natural component of the structure of matter, as well as an engineering component capable of generating a quantum field.
The quantum matrix of Meridian fluctuated, causing it to shift into our dimension for a brief period every sixty years. In 2371, when Jadzia Dax opted to remain on Meridian, the transporter buffer was used to bring her quantum matrix in sync with the planet's during its next shift. However, the planet's matrix began to destabilize due to her presence, forcing Chief O'Brien to beam her back to the USS Defiant. (DS9: "Meridian")
In 2375, the crew of the USS Voyager constructed a quantum slipstream drive, of which a quantum matrix was a central component. In an alternate timeline, while navigating the slipstream, Harry Kim, who was aboard the Delta Flyer piloting ahead of Voyager, sent an erroneous set of phase corrections to Voyager, causing the quantum matrix to overload, resulting in the destruction of the ship. Fifteen years later, in an attempt to change history, Kim utilized a stolen Borg temporal transmitter to send a new set of phase corrections back in time, where they were detected by Seven of Nine's interplexing beacon. As Seven implemented the corrections, the timeline was altered and the slipstream dispersed, returning Voyager safely back to normal space. (VOY: "Timeless")