I'm not really sure, and I'll bet it depends on the form of the sodium. Pure sodium normally reacts with chlorine (in bleach) to make sodium chloride (table salt), but sodium reacts even more strongly with oxygen, which is in the water that the bleach is dissolved in, to make sodium oxide. I doubt you would get anything different for this reason than if you dropped sodium directly into water.

Warning: DON'T try this experiment unless you are wearing protective gear and know what you're doing. The reaction between sodium and water releases a lot of energy. You could be splattered with boiling water, set on fire, breathing sodium oxide vapor, or even blown to bits if you used enough sodium.