There could be many factors here. You can follow below troubleshooting steps.
- Check if firewall is open from F5 to the FQDN node port. Try with telnet comand from F5.
- Check routing is there on F5 and it's not using the mgmt route to reach. tmm route should be present on F5 to reach the FQDN.
- Check the health monitor if any custom http/https monitor configured. Try using tcp for troubleshooting.
- If F5 can connect on fqdn port. Check if node is up or down. disable ICMP monitoring at the node level.
- if FQDN node is in a non default route domain then there is no support for FQDN node currently. But there is a work around to it as per below
https://devcentral.f5.com/s/articles/FQDN-nodes-in-non-default-route-domains?page=1