Yes. When exporting a feature class to another geodatabase or feature class, the existing attachments can be copied to the output.
When running the Feature Class To Geodatabase tool, the attachments are automatically copied to the output geodatabase. When exporting a feature class using the Feature Class To Feature Class tool, the Maintain Attachments environment setting is checked by default to copy the attachments from the input features to the output features. Refer to ArcGIS Pro: Maintain Attachments (Environment setting) for more information.
Ensure the Maintain Attachments environment setting is checked to copy the attachments to the output features.