I haven't seen any other replies so I'll throw in my two cents. If you already have the information you need you may safely ignore this post! :)
WSPR is a low-speed protocol which really doesn't have anything to do with GPS. I assume this is also true for WSJTX.
You could theoretically connect a GPS module that uses serial communications. Arduinos have built-in UARTs just for this purpose, and it's very easy to send and receive serial data through the existing USB serial interface to/from a host computer. However, you don't want to use the USB connection- you probably want a direct serial connection.
Looking at the Arduino Nano schematic shows that the TxD and RxD lines are also brought in parallel to two IO pins, so you can theoretically connect low-level serial lines directly between the two. I haven't tried this myself, though, hence the word "theoretically".
If you wish to check for yourself, the Arduino Nano schematics are available at several places- for example,
.
IMPORTANT: Serial communications sometimes use RS-232 levels. These are NOT compatible with the Arduino pins!
RS-232 involves higher positive voltages and negative voltages which will likely damage an Arduino.
You are looking for a GPS module with so-called "TTL compatible" serial lines.
