Получить свойство Text объекта TextRange , инициализированного по всему FlowDocument :
FlowDocument myFlowDocument = new FlowDocument(); //get your FlowDocument
//put in some (or it already has) text
string inText = "Hello, WPF World!";
TextRange tr = new TextRange(FlowDocument.ContentStart, FlowDocument.ContentEnd);
tr.Text = inText;
//get the current text out of the FlowDocument
TextRange trPrime = new TextRange(FlowDocument.ContentStart, FlowDocument.ContentEnd);
string outText = trPrime.Text;
//now outText == "Hello, WPF World!";
//to get formatting, looks like you would use myFlowDocument.TextEffects