"i look forward to learn how to make it more elegant with map and grep."

You appear to be assuming that using map and/or grep will automatically make your code more elegant. This is not the case.

The code you already have is easy to read (and therefore maintain) and its intent is clear. I can't say the same for these map/grep alternatives:

print $_->[0] for grep {
    @{$_->[1]}
} 
map { 
    [ $_, [ grep { $_ eq $search } @{$hash->{$_}{member}} ] ]
} keys %$hash;
[download]

print $_->[0] for grep {
    $_->[1]
} 
map { 
    [ $_, scalar grep { $_ eq $search } @{$hash->{$_}{member}} ] 
} keys %$hash;
[download]

Furthermore, for loops are often faster. Benchmark to check this with whatever alternative solution(s) you might be considering.

Here's my test code:

#!/usr/bin/env perl -l

use strict;
use warnings;

my $hash = {
    A => {
        member => [qw{q w e}],
    },
    B => {
        member => [qw{a s d}],
    },
    C => {
        member => [qw{q a z}],
    },
};

my $search = 'q';

print 'OP Solution:';

for my $hk (keys %$hash) {
    for my $m (@{$hash->{$hk}{member}}) {
        if ($m eq $search) {
            print $hk;
        }
    } 
}

print 'map/grep Solution 1';

print $_->[0] for grep {
    @{$_->[1]}
}
map {
    [ $_, [ grep { $_ eq $search } @{$hash->{$_}{member}} ] ]
} keys %$hash;


print 'map/grep Solution 2';

print $_->[0] for grep {
    $_->[1]
} 
map { 
    [ $_, scalar grep { $_ eq $search } @{$hash->{$_}{member}} ] 
} keys %$hash;
[download]

Output:

OP Solution:
A
C
map/grep Solution 1
A
C
map/grep Solution 2
A
C
[download]