More accurately, if your current LTE phone has a nano SIM, then you can simply pop it out, put it in your friend's Nexus 5 and it should work. The problem comes if you need a new SIM, or if your current SIM doesn't have LTE service assigned to it. In either of those cases, you'll need to contact Sprint either for a new SIM, or to activate LTE service. At that time, they may fight you, stating that the phone isn't supported. However, if you don't back down, ask for a supervisor, or try another store, eventually you will get a reasonable person.
Worst case, call the customer service line, and say that you got a Nexus 5 from Google, and need them to send you a new SIM card. They don't have to know you didn't actually buy it from Google. Again, it may take talking to a supervisor or even multiple calls, but eventually, you'll succeed.