As Per page 150 of this 6D manual, the 'Spot metering' setting:

... is for metering a specific spot of the subject or scene. The metering is weighted at the center covering approx. 3.5% of the viewfinder area.

Spot metering area diagram from manual.

When the camera is in this metering mode the selected focus point is not taken into consideration.

However, I understand that in 'Evaluative metering' mode, the camera will bias the area around the active focus point considerably, so that may be the mode that you're looking for.

'Evaluative metering' mode

In 'Evaluative metering' mode the camera will split the viewfinder into a grid of of many small sections called zones. (63 in the case of your camera)

63 zone evaluative metering pattern from this article.

Each section is then independently metered for light level and a complex algorithm within the camera decides on the best overall exposure, taking into account which AF point(s) are used, which way up the camera is, and what mode it's in.

In this mode the zones close to the AF point(s) in use, are given a higher priority, in the exposure calculation.