I'm not aware of something like this for Mojolicious, but I guess you can look at how Tk::Carp does its thing and do the same, except for Mojolicious.

If you look at the source of Mail::IMAPClient, you'll see this comment: "#local $SIG{__WARN__} = \&Carp::cluck; #DEBUG" which hints that the author uses $SIG{__WARN__} for this purpose (and Tk::Carp, pointed out by Corion, appears to work the same way). The following seems to work for me, though it's always worth noting that changes to $SIG{__WARN__} are global, so it's possible you might step on another module's toes or that one might step on yours. This is true even with local, unless you know exactly what the call stack looks like. You can of course limit the effects by using local $SIG{__WARN__} only in those blocks where you actually make calls to Mail::IMAPClient, that should help limit your exposure to such potential problems.

use warnings;
use strict;

package Foo {
    use Carp;
    sub bar {
        carp "Foo Bar";
    }
}

use Mojo::Log;

my $log = Mojo::Log->new;
local $SIG{__WARN__} = sub { $log->warn(@_) };

Foo::bar();
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i have seen that but I did not dare to use it (especially with Mojolicious). Thanks++ for clarifying.

You already have some excellent advice. I'd be tempted to try using something like Capture::Tiny in the main program to wrap calls to the external module's code. It allows you to capture only stdout, only stderr, both merged, both separately, tee one or both (capture and send to the original destination), and all of this only for the scope of the coderef you choose.

I have not tested the module with Mojo to be sure it works. I would think it's worth trying, as searching the web for the two together does not turn up any dire warnings.