I figured out a solution to setting the font to double-quotes. I believe I now understand the problem, which had to do with how the search and replace function handles double-quotes from the US Keyboard.
When you enter a double quote in the search field, the keyboard enters a Straight Double. If you look in Keyboard Shortcuts (File=>Preferences=>Keyboard Shortcuts) it shows actions for Apostrophe, Straight Double, Single Left, Single Right, Double Left, Double Right. By default the shortcuts for those characters are not available. Maybe they can be set as user-defined shortcuts, but that's another for another lesson; it's not necessary for proving the solution.
I experimented with different approaches, I entered in ANSI and UTF-8 codes to represent Double Quotes, but none of them worked. I searched for other resources and none seemed to have the answer. Then I experimented with selecting a Double-Left in the document sample, then copying and pasting it into the "Text" field in the "Search/Replace" dialog box on the "Search for:" side. Then on the "Replace with:" side, I checked the "Font" checkbox and set the Font to Times New Roman Regular. I clicked "Search" and the "Replace" and "Replace All" buttons became active. I clicked "Replace All" and all of the Double-Left Quotes changed their font. I repeated the same procedure for the Double-Right Quotes and the task was complete.
So here are the steps with screenshots:
To set Left and Right Double-Quote marks to a different font without changing the font for the whole body text:
1. Select the Text Frame in which you want to change the Double-Left and Double-Right Quote marks:


2. Double click inside the Text Frame, select a Double-Left Quote mark press Ctrl-C to Copy (Optional: right-mouse-click over the selection and select Copy from the pop-up menu).

3. Click the mouse to the upper left corner of the Text frame to place the cursor there.
4. From the Edit menu, select Search/Replace/
5. On the Search for: side, check the Text checkbox and click inside the Text field and Paste the Double-Left Quote mark there (Ctrl-V or right-mouse-click and select Paste from the pop-up menu).
6. On the Replace with: side, check the Font checkbox, click on the drop-down font menu and select Times New Roman Regular (You can select another font, like Georgia or Garamond if you prefer).
7. Click the Search button below. The Replace and Replace All buttons become active.
8. Click Replace to replace the font on the first or next found Double-Left. Click Replace All to replace the font in all instances of the Double-Left Quotes in the Text Frame.

9. Click Close. Your Double-Left Quotes will now appear in the desired font.
10. Repeat Steps 2 through 9, but paste the Double-Right Quotes in the Search For Text field to change the font for Double-Right Quotes.

The key to making this change is getting the correct Double-Quote character into the "Search/Replace=>Search for:" Text field.