For VMware infrastructure:
1. Open Veeam Virtualization Extensions for System Center console:

2. Open Veeam Extension UI > Log in > On the right side click Help > Export Log > Select collector(s) and click Export.

In case, for some reasons, it is impossible or inconvenient to use Veeam UI to export the Veeam logs you can collect the logs manually:
a) Veeam Collector logs
The default location on each Veeam Collector server installed:
C:\Program Files\Veeam\Veeam Virtualization Extensions for System Center\Collector\Log
b) Veeam Virtualization Extension logs
The default location on the server where the Veeam Virtualization Extension service is installed:
C:\Program Files\Veeam\Veeam Virtualization Extensions for System Center\Manager\Log
c) Veeam UI logs
The default location on the server where the Veeam UI component is installed:
C:\Program Files (x86)\Veeam\Veeam Virtualization Extensions for System Center\User Interface\Logs
3. Also, Operations Manager event logs from all (related) Veeam Collector machines and Management Server (where VES installed) might need to investigate the issue:
To export the event logs:
- Operation Manager logs (*evtx) from the Management Server (Event Viewer > Application and Services Logs > Operations Manager > Save All Events As *.evtx).
- Operation Manager logs (*evtx) from all Veeam Collectors (Event Viewer > Application and Services Logs > Operations Manager > Save All Events As *.evtx).
For Hyper-V and Veeam Backup&Replication infrastructure:
Operations Manager event logs from the affected Hyper-V servers and SCOM Management Server / Veeam Backup&Replication Server (where VES installed) might need to investigate the issue:
To export the event logs:
- Operation Manager logs (*evtx) from the Management Server (Event Viewer > Application and Services Logs > Operations Manager > Save All Events As *.evtx).
- Operation Manager logs (*evtx) from all Hyper-V servers / Veeam Backup & Replication servers (Event Viewer > Application and Services Logs > Operations Manager > Save All Events As *.evtx).

For any installation issues:
C:\ProgramData\Veeam\Setup\Temp\Logs\VESSetup.txt