With the increased real-world deployment of quantum computers, there is a security need to be able to fingerprint and track their equipment. This work proposes that cryogenic equipment used in superconducting qubit quantum computers could leverage inexpensive SRAM-based PUFs as fingerprints. This work is the first to perform a security evaluation of SRAM PUFs under cryogenic conditions using liquid nitrogen to rapidly freeze the memories to temperatures approaching -195C (-320F or 77K). This work demonstrates that SRAM PUFs can become more stable under cryogenic conditions. As a result, a possible novel application of the SRAM PUFs is to identify and track quantum computer cryogenic hardware. Other means of fingerprinting quantum computer equipment are also possible, for example, based on the frequency of qubits. The ability to fingerprint quantum computers can be on one hand beneficial, to track the equipment, but on the other detrimental as attackers with access to the fingerprints could identify specific machines. Understanding the benefits and dangers of fingerprinting quantum computers, and securely deploying fingerprinting mechanisms is necessary to protect these emerging computing platforms.