Personally, I do not see any reason why a long company name could not be shortened for convenience within a Quality Plan Document, as long as a reference to the full company name is made somewhere.
One way to look at this is who the audience for the document is.
For External customer/client facing documents, Internet websites, or other publicity material it might be best to use the full name of the company.
For internal employee facing documents or intranet websites, the shortened company name would be ok, after the 1st reference in full.
All the best.
John.