Can I use the brush tool and desaturate just one color?

There is no HSL option explicitly (though there have been Feature Requests for that) but see below  - and also dj_paige's far more succint answer.

One method - works well e.g. with older versions of LrC that lack Range Mask, or perhaps in combination with that - uses the AutoMask brush option.

There are two approaches for doing that: 

When you 'PAINT' (drag the mouse pointer around with the button held down) with AutoMask active, the brush only works onto pixels similar in colour, to wherever you began that brush stroke. This is however not very smooth and proportionate in its results IMO.

Or when you repeatedly 'DAB' (click the mouse button, releasing it immediately) with AutoMask active, again only pixels similar to those under the pointer crosshair are painted onto. But this way you can use a large (or very large) brush feathered to taste, and build up a very effective and smooth selection cumulatively. 

Another option was added in recent versions of Lightroom Classic: a given brush mask, however achieved, can have its application further narrowed by a "Range Mask". Set to Color, an eyedropper can define a small sample area of the photo. Your brush adjustments then only happen onto those brushed parts of the photo, where there is similarity to the colours within this sample area. They are otherwise masked off - they are brushed, but the effect is the same as if they had not been brushed.

So say you wanted to desaturate just the red bits within one area of the photo: you could brush over that whole area of the photo. Then use Range Mask to make your adjustments not apply to the brushed pixels unless they are red. The adjustments being applied, the brushed extent, and the range mask settings all still remain further alterable.

The same technique can also be used with tonal selectivity instead of hue selectivity.

You can use the Color Rank Mask to select just that one color, then desaturate with the Saturation slider.