First of all, ++ for having specified that this is homework in the first place, and for showing what you have tried thus far. Said this, I personally believe that your program is more complex than it ought to be. For example the following may suffice:

#!/user/bin/perl

use strict;
use warnings;

print "Welcome to the mathq program\n";

while (1) {
    print "Enter a number\n";
    chomp(my $n = <STDIN>);
    die "Goodbye\n" if $n eq 'q';
    chomp(my $m = <STDIN>);
    print "$n + $m = ", $n+$m, "\n",
          "$n - $m = ", $n-$m, "\n",
          "$n x $m = ", $n*$m, "\n",
      $m == 0 ? '' :
         ("$n / $m = ", $n/$m), "\n";
}

__END__
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Please note that a value in Perl is either ==0 or !=0 -possibly passing through a complaint if warnings are on- and thus you "validation" for input is not going to work. You may want to add your own validation, e.g.

#!/user/bin/perl

use strict;
use warnings;

print "Welcome to the mathq program\n";

INPUT: while (1) {
    print "Enter a number\n";
    chomp(my $n = <STDIN>);
    die "Goodbye\n" if $n eq 'q';
    chomp(my $m = <STDIN>);
    for ($n,$m) {
        warn "Invalid input: Type a number or 'q' to quit\n" and
          next INPUT unless /\A\d+\z/;
        # use your own more refined validation there
    }
    print "$n + $m = ", $n+$m, "\n",
          "$n - $m = ", $n-$m, "\n",
          "$n x $m = ", $n*$m, "\n",
      $m == 0 ? '' :
         ("$n / $m = ", $n/$m), "\n";
}

__END__
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Or at least force $n and $m to their numeric values, to avoid printing strange things:

    ($n,$m) = map {
        ;no warnings 'numeric';
        $_+0;
    } $n, $m;
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