A1 certificate is not needed for business trip, however it is possible that some countries would request it if employee is traveling there very often, but this can be discussed on case by case basis.
In addition to other business trip expenses, travel allowances are obligatory in Slovenia.
However, please be aware that they might require an A1 certificate which usually costs Eurofast 100 EUR/hourly rate, per diems and travel allowance (more info in the country guide).
Partner: Eurofast
Country-specific or partner-specific: Both
Source: SF00212120