Is it possible to extract specific sentences with same certain format and save them to a text file in Microsoft Word?

You can use VBA to automate the process.

1. Open the document.
2. Press Alt+F11 to open the VBA Editor.
3. Copy and paste the code below.
4. With your cursor inside the code, press F5 to run it. A new window will be opened with the extracted dialogues.

Sub GetDialogues()

    Dim coll As New Collection
    Dim regEx As RegExp
    Dim allMatches As MatchCollection
    
    Set regEx = New RegExp
    
    With regEx
        .IgnoreCase = False
        .MultiLine = True
        .Global = True    'Look for all matches
        .Pattern = """.+?"""    'Pattern to look for
    End With
 
    Set allMatches = regEx.Execute(ActiveDocument.Content.Text)
 
    For Each Item In allMatches
        coll.Add Item   'Add found items to the collection
    Next Item

    Dim newdoc As Document
    Set newdoc = Documents.Add  'Add a new Word document
    newdoc.Activate             'Activate the document

    For Each Item In coll
        newdoc.Content.Text = newdoc.Content.Text + Item   'Add each item (quote) to the document
    Next Item

    newdoc.SaveAs FileName:="test.txt", Fileformat:=wdFormatPlainText   'Save the document as plain text

End Sub