It's not possible to do this with 'live' text - well, other than pasting something on top of it. In that, InDesign is similar to both Illustrator and Photoshop.
For plain text I'd hesitate to suggest it, but for a logo it should be perfectly fine to convert the text to outlines. After you do this, you can select the 'insides' with the white arrow and delete them (make sure not to leave a single point! InDesign can be quite freaky on that!). If you want to give the inside another color, ungroup as many times as needed - text converted to outlines may consist of multiple nested groups - and finally choose "Release Compound Path" on individual objects. That way you get a larger black object - the outside -, and one or more smaller "inside" objects - what originally were the holes. Then you can select each of the holes and give them any color you want.
If you do this, make life easier for your (future) self: select all individual objects again and group them. That way, the chance of accidentally moving a single part away from the rest is a lot smaller.
In InDesign, just select the text and (on OSX) Cmd-Alt-Shift-O then Cmd-Alt-Shift-!
[That keeps the text editable as saved!]
If this logo is a highres bitmap image (600 to 1200 ppi or more in effective resolution when placed on the page), I'd suggest to open it in PhotoShop and transform it to grayscale and then to bitmap (1bit), save as tif and you can color it in InDesign like you want, if selected by Direct Selection Tool or when selected inside its container frame. The now white area is automatically no more white.In fact it is none existent.