Quantitative nucleic acid amplification tests (qNAATs) are critical in treating infectious diseases, such as in HIV viral load monitoring or SARS-CoV-2 testing, in which viral load indicates viral suppression or infectivity. Quantitative PCR is the gold standard tool for qNAATs; however, there is a need to develop point-of-care (POC) qNAATs to manage infectious diseases in outpatient clinics, low- and middle-income countries, and the home. Isothermal amplification methods are an emerging tool for POC NAATs as an alternative to traditional PCR-based workflows. Previous works have focused on relating isothermal amplification bulk fluorescence signals to input copies of target nucleic acids for sample quantification with limited success. In this work, we show that recombinase polymerase amplification (RPA) reactions on paper membranes exhibit discrete fluorescent amplification nucleation sites. We demonstrate that the number of nucleation sites can be used to quantify HIV-1 DNA and RNA in less than 20 minutes. An image-analysis algorithm quantifies nucleation sites and determines the input nucleic acid copies in the range of 67-3,000 copies per reaction. We demonstrate a mobile phone-based system for image capture and onboard processing, illustrating that this method may be used at the point-of-care for qNAATs with minimal instrumentation.