Yes. It is possible to save the REST connections in ArcGIS Pro. Unlike ArcMap, there is no specific folder to store REST connections in ArcGIS Pro. The connections in the Catalog pane are not automatically saved when opening a new project.
To save the REST connections in ArcGIS Pro, create a project template, or set the REST connections as favorites in ArcGIS Pro.
Create a project template
Esri recommends creating a project template in ArcGIS Pro to store all the REST connections. This is practical if different teams from an organization require their own customized set of folder and database connections. This reduces the clutter of unnecessary connections, and provides an organized template for the end user. Sharing the project template with current settings creates a customized starting state for ArcGIS Pro, and can be shared to ArcGIS Online organization, Portal for ArcGIS, or saved locally. Refer to ArcGIS Pro: Create a project template for instructions on creating and sharing a project template.
Set project favorites
Project favorites are a collection of frequently used folder, database, and server connections. These favorites are recorded in the roaming profile on the local network and are user specific. In ArcGIS Pro, the Favorites option is used to configure a list of frequently used connections that is displayed on the start page of ArcGIS Pro. Refer to ArcGIS Pro: Project favorites for more information.