# The string these literals produce don't contains any "\".
my $string = "my string with \ and / slashes."; 
my $substring = "with \ and / slashes"; 
print("$string\n");
print("$substring\n");
print("\n");

# This is what the literals should be.
$string = "my string with \\ and / slashes."; 
$substring = "with \\ and / slashes"; 
print("$string\n");
print("$substring\n");
print("\n");

# It works.
$string =~ s/\Q$substring\E/\"$substring\"/gi;
print("$string\n");
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my string with  and / slashes.
with  and / slashes

my string with \ and / slashes.
with \ and / slashes

my string "with \ and / slashes".
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Hi

Check the input itself is not containing the '\', because it is interpolated in double quotes.

use strict;

my $string = "this is my sting with \ and /slashes. put substing in qu
+otes"; 
my $substring = "with \ and /slashes"; 
print $string;

$string =~ s/\Q$substring\E/\"$substring\"/gi;

OUTPUT:
this is my sting with  and /slashes. put substing in quotes
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Instead use single quote in the input like,

use strict;

my $string = 'this is my sting with \ and /slashes. put substing in qu
+otes'; 
my $substring = 'with \ and /slashes'; 
print $string;

$string =~ s/\Q$substring\E/\"$substring\"/gi;

OUTPUT:
this is my sting with \ and /slashes. put substing in quotes
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This will give the correct output

Punitha