Benito Mussolini did adopt and use the SPQR of the Roman Empire as an attempt to recall that early period of Roman greatness. This is not unlike the way Hitler and his German fascists adopted the innocent swastika, and turned it into one of the more hateful symbols of the modern era.

Symbols are not inherently good or bad. The only thing that is either negative or positive is how they are used and viewed. Timing is everything: having a swastika painted on one's barn in Pennsylvania represented nothing more than a charming folk custom in 1920. By 1942, that innocent farmer may have been searching for a can of paint!

As far as I know, Mussolini did not pollute SPQR the way Hitler did with the swastika, but since there is no standard text to consult in these matters, "as far as I know" only extends that far. What you are requesting is a subjective evaluation of the letters, SPQR. Some may view it as your friend does. Since there is no right or wrong about it, the question is then one of prevalence of this negative view and of your tolerance when dealing with those who might view it negatively.

An additional word of advice just in case you're considering the subject of unusual placement: if you do decide to have the tattoo in question, don't put it across your forehead (in case you were thinking about it): that would cause a great many questions and it may affect your ability to get a job. But then, I am a humanist from a previous century, so my thoughts on this matter may be dated.

The city of Rome continues to use SPQR on all municipal property like water mains, manhole covers, dumpsters and the like. So get ready for some weird looks if you decide to get that tattoo and then go to Rome.

I think your friend is mainly making links in his or her own mind. Much like the swastika, the Nazis did rip off a lot of prominent historical symbols in an effort to link themselves to the great societies of the past.

Frequently, you will see SPQR as part of a legion standard below an eagle with extended wings, like this.

In Nazi propaganda videos you will see a standard like this.

That visual is about the only link I can think of. SPQR (Senatus Populusque Romanus) loosely translates to "The Senate and Roman People". Not sure that or the hundreds of other ways its represented in state and military insignia are in anyway linked to Nazis.