... m[10-KQ^/*$] ...

Huh?!? You really need to start using  <code> ... </code> tags. Please see Markup in the Monastery, Writeup Formatting Tips, How do I post a question effectively?.

How would I incorporate the code [BrowserUk] suggested ...

One way would be to copy the contents of the  m// given by BrowserUk into a  qr// regex object as you originally had it, and then interpolate the  qr// into a  m// match as in your OP (I'm using  \z in place of  $ to match absolute end-of-string):

c:\@Work\Perl\monks>perl -wMstrict -le
"my @tests = qw[
   10-K  10-KSB  10-K405  10-KSB405  10-Q
   10-K/B  10-KA/  10-KSB/ABC  10-K405/A  10-KSB405/A  10-Q/A
   ];
 ;;
 my $formget = qr{ 10- [KQ] [^/]* \z }xms;
 ;;
 print qq{'$_': }, $_ =~ m[^$formget] ? 'matches' : 'does not match' f
+or @tests;
"
'10-K': matches
'10-KSB': matches
'10-K405': matches
'10-KSB405': matches
'10-Q': matches
'10-K/B': does not match
'10-KA/': does not match
'10-KSB/ABC': does not match
'10-K405/A': does not match
'10-KSB405/A': does not match
'10-Q/A': does not match
[download]

Please see perlre, perlrequick and perlretut.

Also: What version of Perl are you working with? It may be useful to know this for future reference.

Update: I forgot about case insensitivity. You can add this with an  /i modifier:
    my $formget = qr{ 10- [KQ] [^/]* \z }xmsi;
or, better, IMHO, for stylistic reasons,  (?i)
    my $formget = qr{ (?i) 10- [KQ] [^/]* \z }xms;
or, best of all,
    my $formget = qr{ 10- [KkQq] [^/]* \z }xms;
because it avoids the  /i performance hit if you are processing very many strings or very long strings.