If you upload your document to Atticus, it will maintain the quotes as formatted in your Word document.
When typing in Atticus, however, be mindful that straight quotes will be used. You can manually apply curly quotes using Keyboard Shortcuts or use the Apply Curly Quote option in the Right Sidebar to automatically add curly quotes.
Since no algorithm can apply curly quotes with 100% accuracy, be sure to check your document and make manual changes where necessary.