No, it will not damage the SSD.

Does chkdsk damage SSD?

The short answer is yes. Running CHKDSK will not harm an SSD in the same way that running DEFRAG might. The two utilities perform differently on an SSD.

How do I check my SSD for errors?

Check HDD/SSD failures with chkdsk
Boot into your Windows.Click Start.Go to Computer.Right-click on the main drive you want to check.Click Properties.At the Tools tab, click Get started at the Error-checking section.Check the Automatically fix file system errors checkbox.Click Start.

When You Should Use SFC

Whereas CHKDSK finds and fixes errors in the file system of your hard drive, SFC (System File Checker) specifically scans and repairs Windows system files. Knowing when to use SFC is usually more obvious than with CHKDSK, which depends on a hunch that your hard drive isn’t behaving correctly.