WQ1 is the spreadsheet format used by Quattro Pro for DOS, versions 1-4.
See also Wikipedia:Quattro Pro.
Modern software that reads WQ1 files
As of 2014, the only modern software that reads WQ1 files is Quattro Pro (part of the WordPerfect Office suite), although with some limitations. Older versions of Microsoft Excel (2003 and earlier) were able to read the format, but support for the Quattro Pro formats was removed altogether in Excel 2010. LibreOffice and OpenOffice don't support the format either. See this blog post for more details.