You will need to if you have either a two-way Security Trust Relationship set at the Active Directory Server, or you will need to configure the same User Name and Password on both domains connect from 1 AD Network to another AD Network (Workgroup authentication).
The issue with your scenario here is that the Solarwinds Alerting Service is the account that is configured by default to send out the execute a program task. You will need to configure the account that exists on both domain to have admin access to the Solarwinds Server. Go into Services.msc, edit the Solarwinds Alerting Service, and change the Log On to the the 2 domain account. Other than permissions, I don't see an issue with using an AD account on the Alerting Service.