Yes. In Lightroom, use a local adjustment rather than a global color change. The usual approach is to apply an Adjustment Brush (or similar local mask) to the background only, then reduce saturation, brightness, or other settings there.

To improve precision, enable Auto Mask and adjust the brush size and feather so Lightroom follows edges more carefully. If you paint over the subject by mistake, use the erase/subtract option to remove that part of the mask.

That said, fine details like hair are difficult to isolate perfectly in Lightroom. If the background and hair share similar colors, Lightroom may still affect both, especially around edges. For very precise work—particularly around hair—Photoshop with layer masks is generally the better tool.