I want to know if it were possible to integrate models created with Blender into a game written in solely C++.
The programming language doesn't matter for this purpose. You just need to get your model data from Blender into your application, which is possible by using one of the various export formats that Blender offers.
Of course all models you create in Blender must be suitable for real-time rendering, i.e. their polygon count should be decent and their materials and other effects features must be supported by your game framework. Typically you'll need many iterations until a content pipeline becomes actually stable.
See this question for more information on suitable formats for use with Blender 3D.
For Blender you can use Python to write script for exporting models to your model data type.