There is only one file type that uses the .KLQ extension:
Kaspersky Quarantine File
Quarantine file created by Kaspersky software products, such as Anti-Virus and Internet Security; contains viruses encrypted and quarantined by Kaspersky products.
KLQ files can become rather sizeable over time so one may wish to delete them. In order to remove KLQ files, 'Self-Defense' must be disabled in Kaspersky Internet Security → Settings → Options → Self-Defense. After disabling 'Self-Defense', the files can be deleted, but ensure to enable 'Self-Defense' again.
Updated: April 3, 2015