How can one use the SPI hardware on the Arduino Nano?
Yes, the Nano supports SPI when used with a modern Arduino version, and no, you do not have to do anything special to utilize it on this board.

The information you found that seems to suggest something to the contrary is simply a result of the fact that that the web page for the Nano is sadly out of date date and does not reflect the current state of the Arduino libraries which support SPI on this processor and hence board.

In terms of components, the Nano it is very nearly just a form-factor shrink of the discontinued Duemilanove, and in the current version updated to an an ATmega328p it has capabilities essentially comparable to an Uno, differing primarily the in the USB interface which has nothing to do with SPI.

In terms of what is wrong with the web page, if we examine archive.org's records for the page of the comparable Duemilanvoe, on September 15th, 2010 it said:


  SPI: 10 (SS), 11 (MOSI), 12 (MISO), 13 (SCK). These pins support SPI communication, which, although provided by the underlying hardware, is not currently included in the Arduino language.


But a few days later on September 26th, 2010 it had been changed to say:


  SPI: 10 (SS), 11 (MOSI), 12 (MISO), 13 (SCK). These pins support SPI communication using the SPI library.


Although the page for that once-flagship product was updated to reflect SPI support nearly six years ago, they've not yet made a corresponding correction to the Nano page.  But as the difference is in the software and the same software is used in either case, the addition of SPI support to the Duemilanove was also automatically an addition of that support to the Nano.