The option to send Pihole-logs to a remote syslogserver is not implemented.
Good thing is that Pi-hole creates log-files.
How about adding these logfiles to the local (R-)Syslog-daemon and send it over to the external Syslog-Server via port 514 TCP/UDP ?

I use Dietpi as the OS on my Raspi3+ and added Pi-hole as one of the supported apps .

Dietpi has per default noSyslog-daemon activated.
The first step is to install a Syslog-daemon, in my case Rsyslog.
Install with root-privileges (sudo su):

apt install rsyslog

cd to /etc/rsyslog.d and create two files:

nano pihole.conf

$InputFileName /var/log/pihole.log
$InputFileTag pihole
$InputRunFileMonitor
$InputFilePersistStateInterval 1000

nano piholeftl.conf

$InputFileName /var/log/pihole-FTL.log
$InputFileTag pihole-ftl
$InputRunFileMonitor
$InputFilePersistStateInterval 1000

as next alter rsyslog.conf and add the following lines

nano /etc/rsyslog.conf

$ModLoad imfile
$InputFileName /var/log/pihole.log
$InputFileName /var/log/pihole-FTL.log
*.* @10.50.100.5:514

10.50.100.5 is the external Syslog-Server. Dietpi sends now the syslog-information to the Syslog-Server 10.50.100.5 via TCP port 514.
Change the IP 10.50.100.5 to the Syslog-Server IP you want to use.

restart Rsyslog-daemon:

systemctl restart rsyslog

The Syslog-daemon receives now the Pi-hole logs:

Addition: If you want to see also the DNS-queries go to Settings and Enable query logging:

This brings also the DNS content to the external Syslog-Server:

May 14 09:58:27 DietPi local0 pihole May 14 09:58:26 dnsmasq[15529]: query[A] bier.de from 10.50.100.13
May 14 09:58:27 DietPi local0 pihole May 14 09:58:26 dnsmasq[15529]: forwarded bier.de to 1.0.0.1
May 14 09:58:27 DietPi local0 pihole May 14 09:58:26 dnsmasq[15529]: dnssec-query[DS] bier.de to 1.0.0.1
May 14 09:58:27 DietPi local0 pihole May 14 09:58:26 dnsmasq[15529]: reply bier.de is no DS
May 14 09:58:27 DietPi local0 pihole May 14 09:58:26 dnsmasq[15529]: reply bier.de is 212.53.128.75
May 14 09:58:27 DietPi local0 pihole May 14 09:58:26 dnsmasq[15529]: query[AAAA] bier.de from 10.50.100.13
May 14 09:58:27 DietPi local0 pihole May 14 09:58:26 dnsmasq[15529]: forwarded bier.de to 1.0.0.1
May 14 09:58:27 DietPi local0 pihole May 14 09:58:26 dnsmasq[15529]: reply bier.de is NODATA-IPv6

Done!